Paper Mill Effluent, Pigouvian Tax, and Permit Cap
Upstream Paper discharges untreated effluent into a river. The mill's marginal private cost of production is MPC = 20 + Q dollars per ton, where Q is tons per day. The market demand for paper is P = 100 - Q. Each ton of paper produced imposes an external cost of $10, so marginal external cost is constant at MEC = 10; assume no other distortions. The mill currently operates in a competitive market and ignores the external cost. The state environmental agency is considering a Pigouvian tax per ton, a cap-and-trade permit system set at the socially optimal quantity, and a direct standard that would require the mill to cut output by 5 tons per day. The mill argues that any tax will raise paper prices, cut output, and cost local jobs. The agency wants to know the size of the tax, the revenue it would raise, the permit price under a cap, and how the policies differ if abatement costs are uncertain.
1. Find the competitive equilibrium price and quantity and the socially optimal price and quantity. What is the deadweight loss from the externality?
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Competitive equilibrium: set MPC equal to demand, 20 + Q = 100 - Q, so 2Q = 80 and Q = 40; P = 100 - 40 = $60. Social optimum: add marginal external cost, MSC = MPC + MEC = 30 + Q. Set 30 + Q = 100 - Q, so 2Q = 70 and Q = 35; P = $65. The unregulated market overproduces by 5 tons per day. The deadweight loss is the triangle between the demand curve and the MSC curve from Q = 35 to Q = 40. At Q = 40, MSC = $70 and demand price = $60, so the gap is $10; at Q = 35 the gap is zero. DWL = 1/2 x 5 x 10 = $25 per day. This is the value of the excess paper minus its true social cost.
2. Design a Pigouvian tax per ton that achieves the social optimum. How much revenue does it raise? If instead the state caps output at 35 tons per day with tradable permits, what is the permit price likely to be, and how does the tax compare with a cap when abatement costs are uncertain?
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A Pigouvian tax equal to marginal external cost at the optimum, $10 per ton, makes the mill's private cost 20 + Q + 10 = 30 + Q. Setting this equal to demand gives 30 + Q = 100 - Q, Q = 35, the social optimum. Tax revenue is $10 x 35 = $350 per day. A cap of 35 tons per day with tradable permits would also achieve Q = 35. If the mill is the only firm and its marginal abatement cost at 35 tons is $10 per ton, the permit price would be about $10; if permits are auctioned, the mill would pay up to its marginal abatement cost. Under uncertainty, a tax fixes the price of emissions and lets quantity vary, while a cap fixes quantity and lets the permit price vary. If abatement costs are higher than expected, the tax produces less abatement and more pollution; the cap holds pollution at 35 but permit prices rise. If costs are lower, the tax produces more abatement; the cap holds quantity and permit prices fall.
3. The mill says the tax will cut output and jobs, so the state should not impose it. Evaluate this claim, including whether the tax revenue could be used to address the job concern.
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The tax does raise the price from $60 to $65 and reduce output from 40 to 35 tons per day. That is not a side effect to be dismissed; it is the correction. At Q = 40, each ton costs society $10 in external damage that the mill does not pay, so the market price understates true social cost. The tax forces the mill to internalize that cost. The revenue is $350 per day, which the state could use to retrain workers, compensate affected households, or cut other taxes; the efficiency gain includes the $25 deadweight loss plus the reduction in external damage, but the distribution of gains and losses is a separate question. Job losses in paper production may be partly offset by jobs in cleanup or other sectors. The claim that 'any tax costs jobs' is too broad: a tax can improve efficiency while creating transition costs, and the policy design-phase-in, revenue use-determines who bears them.