Signature
Q_p = (a - c) / b; d_0 = m - g * Q_p; dQ = s / b; S_g = d_0 * dQ - 0.5 * (b + g) * dQ^2; W = S_g - lambda * s * (Q_p + dQ); W_T = S_g - lambda * s * dQ
| Inputs | Definition | Unit |
|---|---|---|
a | Marginal private benefit of the first unit | pounds per unit |
c | Constant marginal cost of a unit, also its unsubsidised price | pounds per unit |
b | Fall in marginal private benefit per thousand units, above zero | pounds per unit per thousand units |
m | Marginal external benefit of the first unit | pounds per unit |
g | Fall in marginal external benefit per thousand units, zero or above | pounds per unit per thousand units |
s | Subsidy paid on each unit, zero or above | pounds per unit |
lambda | Deadweight loss of raising one more pound of tax revenue, as in HE-FM-DWL-004 | pounds per pound of revenue |
Q_p | Uptake at which marginal private benefit equals marginal cost | thousand units a year |
|---|---|---|
d_0 | Marginal external benefit at Q_p, equal to the gap between marginal social benefit and marginal cost there | pounds per unit |
dQ | Rise in uptake when the price falls by s | thousand units a year |
S_g | Area between marginal social benefit and marginal cost over the dQ units above Q_p | pounds a year, thousands in the article's example |
W | Welfare gained from the induced units less the excess burden of the whole bill | pounds a year, thousands in the article's example |
W_T | Welfare gained from the induced units less the excess burden of the bill on those units only | pounds a year, thousands in the article's example |
Function
Pigouvian subsidy bill, inframarginal transfer and net welfare with costly public funds
Maps linear marginal private benefit, constant marginal cost and falling marginal external benefit curves, a per-unit subsidy and the marginal excess burden of taxation to the subsidy bill, the part of it paid to people who would have taken up the activity anyway, the net welfare change of the subsidy and the uniform rate that maximises it. The efficient quantity and the Pigouvian subsidy s = MEB(Q_s) with these curve shapes are HE-FM-EXT-002, within the social marginal benefit framework HE-FM-EXT-001, so the article's efficient 72 thousand courses, its subsidy of GBP 12 and its welfare loss of GBP 90,000 a year without a subsidy are re-applications of HE-FM-EXT-002 and are not drafted again. The subsidy set at the external benefit observed before the subsidy has the same algebra as the tax record HE-FM-PIGT-003 with the sign of the external term reversed (an external cost of minus 30 plus 0.25Q for the article's curves), which returns the article's 75 thousand courses and loss of 5.625. HE-FM-PIGT-003 is written for an external cost of zero or above (HE-VR-PIGT-025, HE-AS-PIGT-006, HE-VT-PIGT-006), and for a subsidy the overshoot lies above the efficient quantity, so the package reads that case through HE-FM-PIGS-002 at a rate equal to the external benefit at Q_p (HE-EX-PIGS-006) and does not draft it again. The excess burden of financing is the marginal excess burden of HE-FM-DWL-004 applied to the subsidy bill. The new steps are the bill and its split between inframarginal and induced users (HE-FM-PIGS-001), the net welfare of any uniform or targeted rate (HE-FM-PIGS-002) and the welfare-maximising uniform rate (HE-FM-PIGS-003). Notation follows the Pigouvian Subsidy article.
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Implementations
Excel
Net welfare of a Pigouvian subsidy from named curve cells
With the curve parameters in DemandIntercept, DemandSlope, MargCost, ExtIntercept and ExtSlope, the marginal excess burden in ExcessBurden and the rate in SubsidyRate, the formulas fill PrivateQty, MebAtPrivateQty, InducedUnits, SurplusGain, NetWelfareUniform and NetWelfareTargeted in turn.
=(DemandIntercept-MargCost)/DemandSlope; =ExtIntercept-ExtSlope*PrivateQty; =SubsidyRate/DemandSlope; =MebAtPrivateQty*InducedUnits-0.5*(DemandSlope+ExtSlope)*InducedUnits^2; =SurplusGain-ExcessBurden*SubsidyRate*(PrivateQty+InducedUnits); =SurplusGain-ExcessBurden*SubsidyRate*InducedUnits
Assumptions
Linear curves and full pass-through of the subsidy
Marginal private and external benefit are linear and marginal cost constant over the range of uptake between Q_p and Q_p plus dQ, as in HE-AS-EXT-004, and the subsidy lowers the price people face by s, as in HE-AS-EXT-006. Goodkin-Gold and colleagues rely on the same pass-through of a subsidy by competitive firms. Their steady-state external benefit of a vaccine rises with coverage, so the falling linear benefit here is an illustration, as the article says.
One marginal excess burden on every pound of the bill
Each pound of the bill is raised at the same marginal excess burden lambda, the assumption of HE-AS-DWL-010, and the bill itself is a transfer. OMB Circular A-94 sets 25 cents per dollar of revenue for a supplementary analysis, the article uses an illustrative 20p, and lambda of zero corresponds to Pigou's proviso that the funds be raised by a mere transfer.
Targeting that is exact and costless for the targeted net welfare
W_T supposes that the subsidy reaches only the people induced to take up the activity, at no cost of targeting, as in the article's case of a subsidy confined to the extra 12 thousand courses. It is a bound on what targeting can achieve; Goodkin-Gold and colleagues argue for subsidies targeted even very imprecisely to people or areas with low take-up.
Worked examples
Pigouvian rate of GBP 12 with a 20p excess burden
With the article's illustrative curves MPB = 100 minus Q, MC = 40 and MEB = 30 minus 0.25Q, a uniform subsidy of GBP 12 gains GBP 90,000 a year but the excess burden of the GBP 864,000 bill at 20p a pound is GBP 172,800, a net loss of GBP 82,800. Confined to the 12 thousand extra courses the net gain is GBP 61,200, the article's figures.
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0.2; s = 12; Q_p = 60; d_0 = 15; dQ = 12; S_g = 90; W = -82.8; W_T = 61.2
Uniform subsidy of GBP 1.82 with a 20p excess burden
At the article's best uniform rate of about GBP 1.82 a course the net gain is about GBP 2,700 a year, as in the article; the gain from the 1.82 thousand induced courses is GBP 25,230 and the excess burden of the bill GBP 22,502 (computed here).
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0.2; s = 1.82; Q_p = 60; d_0 = 15; dQ = 1.82; S_g = 25.22975; W = 2.7273; W_T = 24.5673
Subsidy set at the pre-subsidy external benefit of GBP 15 with costless funds
With lambda of zero, a subsidy of GBP 15, the external benefit at 60 thousand courses, raises uptake to 75 thousand and gains GBP 84,375 a year, short of the GBP 90,000 at the Pigouvian rate of HE-FM-EXT-002 by the article's overshoot loss of GBP 5,625.
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0; s = 15; Q_p = 60; d_0 = 15; dQ = 15; S_g = 84.375; W = 84.375; W_T = 84.375
Steeper demand halves the uptake response to a GBP 12 subsidy
With illustrative curves MPB = 100 minus 2Q, MC = 40 and MEB = 30 minus 0.5Q, private uptake is 30 thousand and the Pigouvian rate of HE-FM-EXT-002 is again GBP 12, but it induces only 6 thousand courses. The gain is GBP 45,000 and the excess burden of the GBP 432,000 bill at 20p a pound GBP 86,400, a net loss of GBP 41,400, or a net gain of GBP 30,600 if confined to the induced courses (computed here for illustration).
a = 100; b = 2; c = 40; m = 30; g = 0.5; lambda = 0.2; s = 12; Q_p = 30; d_0 = 15; dQ = 6; S_g = 45; W = -41.4; W_T = 30.6
Common errors
Excess burden charged only on the induced units of a uniform subsidy
A uniform subsidy is paid to inframarginal users too, so its excess burden falls on the whole bill. Charging it only on the induced courses in the article's example reports a net gain of GBP 61,200 when the uniform subsidy of GBP 12 loses GBP 82,800, a gap of GBP 144,000 equal to the excess burden on the GBP 720,000 transfer (computed here for illustration).
Uptake response to a Pigouvian subsidy taken as the rate itself
Uptake rises by s / b, not by s. In the steeper-demand example, with b of 2, writing dQ as 12 in place of 6 gives a welfare gain of zero in place of GBP 45,000 and a net change of minus GBP 100,800 in place of minus GBP 41,400 (computed here for illustration). The two coincide only when b is 1, as in the article's curves.
Sources
Welfare as social benefit less cost and the subsidy equal to the marginal externality
Goodkin-Gold M, Kremer M, Snyder CM, Williams HL. Optimal vaccine subsidies for endemic and epidemic diseases. NBER Working Paper 28085. Cambridge, MA: National Bureau of Economic Research; 2020. Section 3.4, Normative Measures, which writes welfare as W(Q) = SB(Q) minus cQ with marginal social benefit the derivative of SB and the first best solving MSB(Q**) = c, and section 3.7, Government Subsidies, which states that competitive firms pass the subsidy through to consumers and that GS** = MEX(Q**), the familiar result that setting the subsidy equal to the marginal externality is optimal. Section 2.4, also titled Normative Measures, gives in equation (22) a steady-state marginal externality that rises with coverage below Q0 and is zero once enough vaccine is available to eradicate the disease.
OMB marginal excess burden of 25 cents per dollar of revenue
Office of Management and Budget. Circular No. A-94 (Transmittal Memo No. 64). Guidelines and discount rates for benefit-cost analysis of federal programs. Washington, DC: OMB; 1992. Section 11, Special Guidance for Public Investment, which states that taxes impose a burden in excess of the revenues they raise, that a reasonable estimate of the marginal excess burden is 25 cents per dollar of revenue, and that costs in the form of public expenditures are multiplied by 1.25 in a supplementary analysis; the section is not required for cost-effectiveness analyses.
Pigou's proviso on funds raised by a mere transfer
Pigou AC. The Economics of Welfare. 4th ed. London: Macmillan; 1932. First published 1920. Part II, Chapter XI, section 11, which states that where the value of the marginal social net product exceeds that of the marginal private net product, certain rates of bounty increase the national dividend provided that the funds for the bounty can be raised by a mere transfer that does not inflict any indirect injury on production.
Canonical Identity
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