Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Pigouvian tax on a negative externality set at marginal external cost at the efficient quantity

MPB(Q_star) = MPC(Q_star) + MEC(Q_star); t_star = MEC(Q_star)

Finds the quantity at which the marginal private benefit of an activity equals its marginal social cost, the marginal private cost plus the marginal external cost, and sets a tax per unit equal to the marginal external cost at that quantity, so that the decision-maker faces the full social cost of each unit. It is the negative-externality mirror of the vaccination subsidy HE-FM-EXT-002 within the social marginal cost and benefit framework HE-FM-EXT-001 on the Externality page. The formulae below apply the rule with a linear demand curve, a constant marginal private cost and a marginal external cost that rises with the quantity consumed.

  • Pigouvian tax and efficient quantity with linear demand and rising marginal external cost

    Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); t_star = m + g * Q_star

    With marginal private benefit (the demand curve) a minus b × Q, a constant marginal private cost c and a marginal external cost m plus g × Q, consumers left alone choose Q_0, where marginal private benefit equals c. The efficient quantity Q_star sets marginal private benefit equal to marginal social cost, c plus the marginal external cost. The Pigouvian tax t_star is the marginal external cost at Q_star, and adding it to the private cost moves the private choice to Q_star.

  • Deadweight loss of an untaxed negative externality with linear curves

    Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); DWL = 0.5 * (Q_0 - Q_star) * (m + g * Q_0)

    Without a tax, every unit between Q_star and Q_0 costs society more than it is worth to consumers. With linear curves the loss is a triangle: its base is Q_0 minus Q_star, and its height is the gap between marginal social cost and marginal private benefit at Q_0, which equals the marginal external cost at Q_0 because marginal private benefit equals c there. The Pigouvian tax removes the whole triangle.

  • Quantity and remaining welfare loss when the tax is set at the pre-tax marginal external cost

    Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); t_0 = m + g * Q_0; Q_t = (a - c - t_0) / b; L_t = 0.5 * (Q_star - Q_t) * ((a - b * Q_t) - (c + m + g * Q_t))

    Sets the tax t_0 at the marginal external cost observed before the tax, at Q_0, rather than at Q_star. When marginal external cost rises with consumption, t_0 exceeds the Pigouvian rate and consumers cut back to Q_t, below the efficient quantity. The remaining loss L_t is the triangle between Q_t and Q_star, with height equal to marginal private benefit less marginal social cost at Q_t.