Signature
Q_p = (a - c) / b; d_0 = m - g * Q_p; s_opt = max(0, b * (d_0 - lambda * b * Q_p) / (b + g + 2 * lambda * b)); W_opt = max(0, d_0 - lambda * b * Q_p)^2 / (2 * (b + g + 2 * lambda * b))
| 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 |
lambda | Deadweight loss of raising one more pound of tax revenue | 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 | pounds per unit |
s_opt | Uniform per-unit subsidy that maximises net welfare, zero or above | pounds per unit |
W_opt | Net welfare change W at s_opt | 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
Best uniform Pigouvian subsidy from named curve cells
With the curve parameters in DemandIntercept, DemandSlope, MargCost, ExtIntercept and ExtSlope and the marginal excess burden in ExcessBurden, the formulas fill PrivateQty, MebAtPrivateQty, BestUniformSubsidy and BestNetWelfare in turn.
=(DemandIntercept-MargCost)/DemandSlope; =ExtIntercept-ExtSlope*PrivateQty; =MAX(0,DemandSlope*(MebAtPrivateQty-ExcessBurden*DemandSlope*PrivateQty)/(DemandSlope+ExtSlope+2*ExcessBurden*DemandSlope)); =MAX(0,MebAtPrivateQty-ExcessBurden*DemandSlope*PrivateQty)^2/(2*(DemandSlope+ExtSlope+2*ExcessBurden*DemandSlope))
Assumptions
Subsidy constrained to be zero or above
s_opt is floored at zero, so the formula never returns a tax on the activity; Goodkin-Gold and colleagues likewise constrain the subsidy to be nonnegative. When d_0 is at most lambda times b times Q_p the best uniform subsidy is zero and W_opt is zero.
Concave net welfare with an interior optimum
With b above zero, g and lambda zero or above, b + g + 2 lambda b is positive and W is a concave quadratic in s, so the first-order condition gives the maximum. The optimum is read within the range where the linear curves of HE-FM-PIGS-002 hold.
Worked examples
Best uniform treatment subsidy of GBP 1.82 with a 20p excess burden
With the article's illustrative curves and an excess burden of 20p a pound, the best uniform subsidy is 3 / 1.65, about GBP 1.82 a course, and the net gain about GBP 2,700 a year, the article's figures, against the Pigouvian rate of GBP 12.
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0.2; Q_p = 60; d_0 = 15; s_opt = 1.8182; W_opt = 2.7273
Costless public funds return the Pigouvian subsidy of GBP 12
With lambda of zero the best uniform subsidy is GBP 12, the Pigouvian rate of HE-FM-EXT-002 for the article's curves, and the net welfare GBP 90,000 a year, the deadweight loss that subsidy removes, a limiting case.
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0; Q_p = 60; d_0 = 15; s_opt = 12; W_opt = 90
Excess burden of 25 cents per pound leaves no worthwhile uniform subsidy
At the OMB supplementary rate of 25 cents per unit of revenue, applied to the article's curves for illustration, the external benefit of GBP 15 at 60 thousand courses equals 0.25 times 60, so the best uniform subsidy and its net gain are both zero (computed here).
a = 100; b = 1; c = 40; m = 30; g = 0.25; lambda = 0.25; Q_p = 60; d_0 = 15; s_opt = 0; W_opt = 0
Common errors
Pigouvian rate kept when public funds are costly
Setting the uniform subsidy at the external benefit at the efficient quantity, GBP 12 in the article's example, ignores Pigou's proviso. At an excess burden of 20p a pound it turns a net gain of about GBP 2,700 at GBP 1.82 into a net loss of GBP 82,800, as the article shows.
Excess burden charged only on the transfer to inframarginal users
Counting the excess burden on lambda s Q_p alone leaves out the part paid on the induced units, so the derivative loses its 2 lambda b term. For the article's curves the rate becomes 3 / 1.25, GBP 2.40, with a net gain of GBP 2,448 against GBP 2,727 at GBP 1.82 (computed here for illustration).
Sources
Pigou on one optimum rate of bounty under a costless transfer
Pigou AC. The Economics of Welfare. 4th ed. London: Macmillan; 1932. First published 1920. Part II, Chapter XI, section 11, which states that there will be one rate of bounty the granting of which would have the optimum effect, provided that the funds for the bounty can be raised by a mere transfer that does not inflict any indirect injury on production.
Deadweight loss of taxation and frictions in public funds in Goodkin-Gold and colleagues
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 3.7, Government Subsidies, which constrains the subsidy to be nonnegative; section 4.2, Targeting Subsidies, which states that authorities may face frictions in raising or dispersing public funds that make programme expenses a relevant consideration; and section 6, Conclusion, which states that with a monopoly supplier the per-unit subsidy needed for the first best can be prohibitively expensive and that, with even a small deadweight loss of taxation, such a subsidy would fail a cost-benefit test.
OMB exceptions to the 25 per cent excess burden
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.b, Exceptions, which states that a different figure may be used where specific information clearly suggests that the excess burden is lower or higher than 25 percent, with an explanation, and that an investment funded by user charges that function like market prices has an excess burden of zero.
Canonical Identity
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