Negative Externality in Electricity Production
A coal-fired power plant produces electricity and emits pollution. The private marginal cost of producing electricity is PMC = 20 + 2Q, where Q is in megawatt-hours (MWh). The marginal external cost of pollution is MEC = 10 + Q. The demand for electricity is perfectly inelastic at 30 MWh. The government is considering a Pigouvian tax per MWh to correct the externality.
1. Determine the socially optimal quantity of electricity and the Pigouvian tax per unit that would achieve it. Show your calculations.
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Social marginal cost SMC = PMC + MEC = (20 + 2Q) + (10 + Q) = 30 + 3Q. Demand is perfectly inelastic at Q=30, so the socially optimal quantity is 30 MWh (since demand is vertical, quantity is fixed at 30 regardless of price). To find the Pigouvian tax, we need the marginal external cost at Q=30: MEC = 10 + 30 = 40. So the tax should be $40 per MWh. Alternatively, the tax equals the difference between SMC and PMC at Q=30: SMC = 30 + 330 = 120, PMC = 20 + 230 = 80, difference = 40. Thus, a tax of $40 per MWh achieves the socially optimal outcome.
2. If the government imposes a tax of $30 per MWh instead of the optimal tax, what quantity will the firm choose? Calculate the deadweight loss from this suboptimal tax.
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With a tax of $30 per MWh, the firm's effective marginal cost becomes PMC + tax = (20 + 2Q) + 30 = 50 + 2Q. The firm will produce where demand equals this effective marginal cost. Since demand is perfectly inelastic at Q=30, the firm will still produce 30 MWh if it can cover its costs. However, the firm will only produce if the price it receives (which is determined by demand) is at least the effective marginal cost. But with perfectly inelastic demand, the price can be arbitrarily high, so the firm will produce 30 MWh. Wait, if demand is perfectly inelastic, the quantity is fixed at 30 regardless of price, so the tax does not change the quantity. Thus, the quantity remains 30 MWh. The deadweight loss is zero because the quantity is already socially optimal. The tax of $30 is less than the optimal $40, so the firm faces a lower cost but still produces the same quantity. However, the tax revenue is lower, but that is not a deadweight loss. The deadweight loss from the tax itself is zero because quantity doesn't change. But the suboptimal tax means the externality is not fully internalized, but since quantity is fixed, there is no inefficiency. So DWL = 0.