The Factory and the River

Waste flows into a river upstream, and the fishing village downstream pays for it.

A 🏭 factory counts only its own cost. Society's true cost adds the damage done to the river. The market hears only the factory's cost and produces too much. A tax sized to the damage closes the gap. The cost-adding formula is below.

For example, the upstream village dumps trash in the river, so 🧑 kids downstream cannot swim in it. The dumpers pay nothing, so they keep dumping. Charge a fine for each bag of trash, and the dumping shrinks.

MSC=MPC+MDMSC = MPC + MD
τ=MD (at the optimum)\tau = MD \ \text{(at the optimum)}

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Examples & Exercises

Demand is P=14−QP = 14 - Q, the factory's cost is MPC=2+QMPC = 2 + Q, and pollution damage is MD=2QMD = 2Q.

With no tax, how much does the factory make, and how far is that from what society wants?

No tax, damage slope 22 — the gap between market and optimum.
Show proof

  1. The factory sees only its own cost: from 14−Q=2+Q14 - Q = 2 + Q, Q=6Q = 6. The village's damage is not in the factory's books, so it drops out of the math.

  2. Society's true cost is MSC=MPC+MD=2+3QMSC = MPC + MD = 2 + 3Q.

    From 14−Q=2+3Q14 - Q = 2 + 3Q, the social optimum is Q=3Q = 3 — the factory makes exactly double.

    The extra 33 units create a deadweight loss of 1818.

The government wants to tax each unit the factory makes on the same river.

How large should the tax be so the factory stops exactly at the social optimum?

Tax 66 — output lands exactly on the social optimum of 33.
Show proof

  1. The answer is the damage at the optimum itself: at the optimal output 33, MD=2×3=6MD = 2 \times 3 = 6, so the tax is τ=6\tau = 6.

  2. With the tax, the factory's math becomes 14−Q=2+Q+614 - Q = 2 + Q + 6, stopping at Q=3Q = 3. All the tax does is put the cost others were paying into the factory's own books — output reaches the optimum on its own and the deadweight loss vanishes.

1. Intro to Microeconomics