Signature
B = s * Q_u; T = s * Q_p; B_n = s * (Q_u - Q_p); c_n = B / (Q_u - Q_p)
| Inputs | Definition | Unit |
|---|---|---|
s | Subsidy paid on each unit taken up | pounds per unit |
Q_u | Units taken up when the subsidy is paid, greater than Q_p | thousand units a year |
Q_p | Units that would be taken up at the unsubsidised price | thousand units a year |
B | Public money paid out when the subsidy is paid on every unit taken up | pounds a year, thousands in the article's example |
|---|---|---|
T | Part of the bill paid on units that would have been taken up without the subsidy | pounds a year, thousands in the article's example |
B_n | Part of the bill paid on the units the subsidy brings about | pounds a year, thousands in the article's example |
c_n | Whole bill divided by the number of extra units the subsidy brings about | pounds per induced unit |
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
Pigouvian subsidy bill and inframarginal transfer from named cells
With the rate in SubsidyRate and the uptakes without and with the subsidy in PrivateUptake and SubsidisedUptake, the four formulas fill SubsidyBill, InframarginalTransfer, InducedUnitsBill and BillPerInducedUnit in turn.
=SubsidyRate*SubsidisedUptake; =SubsidyRate*PrivateUptake; =SubsidyRate*(SubsidisedUptake-PrivateUptake); =SubsidyBill/(SubsidisedUptake-PrivateUptake)
Assumptions
One rate paid on every unit with no displacement of private uptake
The subsidy is paid at the same rate on every unit, so the Q_p units that people would have taken up anyway attract it as well as the induced units. Inframarginal users keep their uptake unchanged, so T is a pure transfer to them and no private activity is displaced. OMB Circular A-94 asks analyses to identify how far a policy such as a subsidy programme promotes substitutes for activities that would occur without it, and to record either the displaced activities as costs or only the incremental gains as benefits.
Uptake under the subsidy taken from the demand response
Q_u is the uptake when the price people face falls by s, as in HE-AS-EXT-006, so with the linear demand of HE-FM-EXT-002 it equals Q_p plus s / b. Q_u must exceed Q_p for c_n to be defined; a subsidy that induces no extra uptake is a transfer only.
Worked examples
Treatment subsidy of GBP 12 on 72 thousand courses
In the article's illustrative example a subsidy of GBP 12 a course raises uptake by 12 thousand courses a year, to 72 thousand. The bill is GBP 864,000 a year, of which GBP 720,000 goes to people who would have been treated anyway and GBP 144,000 is paid on the 12 thousand extra courses, the article's figures. Each extra course costs the budget GBP 72 (computed here).
s = 12; Q_p = 60; Q_u = 72; B = 864; T = 720; B_n = 144; c_n = 72
Lentils incentive with nine inframarginal children per marginal child
Goodkin-Gold and colleagues suppose an incentive worth USD 10 per child immunised and nine inframarginal children for every marginal child, round numbers chosen for ease of analysis. Per ten children paid, nine would have been immunised anyway, so USD 100 is paid out for each new immunisation induced, their figure.
s = 10; Q_p = 9; Q_u = 10; B = 100; T = 90; B_n = 10; c_n = 100
No inframarginal users leaves the bill per induced unit at the rate
If nobody would take up the activity without the subsidy, every unit is induced, the transfer is zero and the bill per induced unit equals the rate of GBP 12, a limiting case computed here for illustration.
s = 12; Q_p = 0; Q_u = 12; B = 144; T = 0; B_n = 144; c_n = 12
Common errors
Pigouvian subsidy bill counted as a cost to society
Subtracting the whole bill from the welfare gain treats a transfer as a resource cost. In the article's example the GBP 864,000 bill against the GBP 90,000 gain gives a net loss of GBP 774,000, when the social cost of raising the bill at the illustrative 20p excess burden is GBP 172,800 and the net loss GBP 82,800 (computed here for illustration). OMB Circular A-94 excludes pure transfers from net present value and asks for their distributional effects to be discussed.
Subsidy rate read as the budget cost of each extra unit
The rate understates what the budget pays for each unit the subsidy brings about whenever most users are inframarginal. In the article's example GBP 12 a course buys 12 thousand extra courses for GBP 864,000, which is GBP 72 per extra course, six times the rate, because 60 of every 72 subsidised courses would have been taken up anyway (computed here for illustration).
Sources
Inframarginal consumers and the lentils calibration of 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 4.2, Targeting Subsidies, which states that a uniform subsidy to induce more high-cost consumers to vaccinate could require paying a high subsidy to inframarginal consumers, amplifying programme expense, and supposes USD 10 worth of lentils per child immunised with nine inframarginal children receiving the subsidy for every marginal child, so that USD 100 of subsidy must be paid out for each new immunisation induced.
OMB Circular A-94 on transfers and incremental gains from a subsidy programme
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 6.a(1), Incremental Benefits and Costs, which asks analyses to identify the extent to which a policy such as a subsidy programme promotes substitutes for activities of a similar nature that would occur without it, and to record either the displaced activities as costs or only incremental gains as benefits, and section 6.a(4), Transfers, which states that there are no economic gains from a pure transfer payment because the benefits to those who receive it are matched by the costs borne by those who pay for it, so that transfers are excluded from net present value and their distributional effects discussed.
Canonical Identity
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