Externalities and the Pigouvian Tax

The factory's cost and the neighborhood's cost come apart here.

The factory counts only its own cost. It never counts the harm from its smoke. Social cost adds the two together, as in the formula below. So the factory produces too much, thinking it is cheap. The lost welfare is the red triangle. Flip the Pigouvian tax on, and the cost curve aims at the social optimum.

For example, a 🏭 factory makes one toy for 100 won. Its chimney smoke dirties the neighbor's laundry. That damage is worth 30 won. The true cost is 130 won, but the factory counts only 100. So it makes too many toys.

MSC(Q)=MPC(Q)+MD(Q)MSC(Q) = MPC(Q) + MD(Q)
τPigou=MD(Qs),Q(τPigou)=Qs\tau^{Pigou} = MD(Q_s), \qquad Q(\tau^{Pigou}) = Q_s

Part of these learning paths

Examples & Exercises

Let demand be P=10−QP = 10 - Q, private marginal cost MPC=2+QMPC = 2 + Q, and marginal damage MD=2+0.5QMD = 2 + 0.5Q.

Find the market equilibrium and the social optimum, and verify by hand that the Pigouvian tax restores the social optimum.

MSC(Q)=MPC(Q)+MD(Q)=4+1.5QMSC(Q) = MPC(Q) + MD(Q) = 4 + 1.5Q
τ=MD(Qs)\tau = MD(Q_s)
Pigouvian tax τ=3.2\tau = 3.2 applied — the dashed curve hits the social optimum.
Show proof

  1. The market equilibrium solves MPC=MPC = demand: 2+Q=10−Q2 + Q = 10 - Q, so Qp=4Q_p = 4. The social optimum solves MSC=MSC = demand: 4+1.5Q=10−Q4 + 1.5Q = 10 - Q, so 2.5Q=62.5Q = 6 and Qs=2.4Q_s = 2.4.

    The market overproduces.

  2. The Pigouvian tax must equal marginal damage at the social optimum: τ=MD(2.4)=2+0.5⋅2.4=3.2\tau = MD(2.4) = 2 + 0.5 \cdot 2.4 = 3.2. Equating tax-inclusive private cost to demand, 5.2+Q=10−Q5.2 + Q = 10 - Q gives Q=2.4=QsQ = 2.4 = Q_s — exactly the social optimum.

In the same example, compute the deadweight loss of the uncorrected externality as a triangle area.

How does the answer change if marginal damage is constant (F=0F = 0)?

No tax — the red triangle is the DWL of 3.23.2.
Show proof

  1. The DWL triangle runs from Qs=2.4Q_s = 2.4 to Qp=4Q_p = 4, between the demand and MSC curves. Its height is the gap at QpQ_p, MD(4)=2+2=4MD(4) = 2 + 2 = 4, and its base is 4−2.4=1.64 - 2.4 = 1.6, so the area is 4⋅1.6/2=3.24 \cdot 1.6 / 2 = 3.2.

  2. With F=0F = 0, MD=2MD = 2 is constant. Then Qs=(10−2−2)/2=3Q_s = (10-2-2)/2 = 3; the triangle has height 22 and base 4−3=14-3 = 1, so DWL is 11.

    With constant damage the Pigouvian tax is also constant at τ=2\tau = 2.

    DWL=MD⋅(Qp−Qs)2DWL = \frac{MD \cdot (Q_p - Q_s)}{2}

10. Externalities and Public Goods