Functions & Formulae

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Pigouvian subsidy bill, inframarginal transfer and net welfare with costly public funds

p(a, b, c, m, g, s, lambda) = (B, T, W, s_opt)

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.

  • Pigouvian subsidy bill, transfer to inframarginal users and bill per induced unit

    B = s * Q_u; T = s * Q_p; B_n = s * (Q_u - Q_p); c_n = B / (Q_u - Q_p)

    Multiplies the per-unit subsidy by the uptake it produces to give the bill to the public budget, and by the uptake there would have been without it to give the transfer to inframarginal users, people paid for doing what they would have done anyway. The rest of the bill is paid on the induced units, and dividing the whole bill by the induced units gives the public money paid out for each extra unit the subsidy brings about. The bill is a transfer and not a cost to society in itself; the social cost of raising it is the excess burden of HE-FM-DWL-004 with the bill as G, which HE-FM-PIGS-002 carries into the net welfare change. With linear demand the uptake under the subsidy is the private quantity of HE-FM-EXT-002 plus s / b.

  • Net welfare change of a uniform or targeted Pigouvian subsidy with costly public funds

    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

    Measures the welfare gained as a per-unit subsidy s raises uptake from the private quantity Q_p of HE-FM-EXT-002 by s / b, the area between marginal social benefit and marginal cost over the induced units, and subtracts the excess burden of raising the bill, lambda times the bill of HE-FM-PIGS-001. W charges the excess burden on the whole bill of a uniform subsidy and W_T on a subsidy confined to the induced units. With the article's curves W reduces to the article's 15s minus 0.625s squared minus 0.2s(60 + s). At any rate S_g equals the deadweight loss of HE-FM-EXT-002 less the triangle of HE-FM-DWL-001 left between Q_p plus dQ and the efficient quantity, so at the Pigouvian rate it equals that deadweight loss and at a rate of d_0 it falls short by the overshoot loss (derived here and checked on a grid of curves). The general form in a, b, c, m and g is derived here from the article's example.

  • Welfare-maximising uniform Pigouvian subsidy when public funds are costly

    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))

    Finds the uniform per-unit subsidy that maximises the net welfare W of HE-FM-PIGS-002 and the net welfare it gives. Setting the derivative of W to zero gives s_opt, which falls below the Pigouvian rate of HE-FM-EXT-002 whenever lambda is above zero and d_0 is positive, because each extra pound of rate is paid on every inframarginal unit as well as on the induced ones. With lambda of zero and a positive d_0, s_opt equals the Pigouvian rate and W_opt the deadweight loss of HE-FM-EXT-002. A subsidy adds net welfare only when the external benefit at the private quantity exceeds lambda times b times Q_p; otherwise the formula returns zero. The closed form is derived here from W and checked against the article's figures.

Pigouvian Subsidy — Functions & Formulae | HealthEconomics.wiki