Reading the scenario
Take an illustrative market for an industrial chemical whose production pollutes a river. All figures below are ETG teaching figures, not from any exam paper. The market demand and the firms' private supply are given by:
- Demand (marginal private benefit = marginal social benefit): P = 60 - 0.5Q
- Private supply (marginal private cost, MPC): P = 10 + 0.5Q
- Marginal external cost of production: a constant $20 per unit
Price P is in dollars per tonne and quantity Q is in thousands of tonnes per year. Because the only spillover is on the cost side, marginal private benefit and marginal social benefit coincide along the demand curve, while the true marginal social cost lies $20 above the MPC.
Step 1: the free-market equilibrium
The unregulated market clears where demand meets MPC. Set the two expressions equal:
60 - 0.5Q = 10 + 0.5Q
60 - 10 = 0.5Q + 0.5Q
50 = 1Q, so Q = 50 thousand tonnes.
Substitute back into demand to find the price: P = 60 - 0.5(50) = 60 - 25 = $35 per tonne. So the free market produces 50 thousand tonnes at $35.
Step 2: the marginal social cost line and the social optimum
Add the $20 external cost to the MPC to get the marginal social cost:
MSC = MPC + external cost = (10 + 0.5Q) + 20 = 30 + 0.5Q.
The socially efficient output is where MSC meets demand (MSB), because there society's extra cost of the last unit just equals its extra benefit:
60 - 0.5Q = 30 + 0.5Q
60 - 30 = 1Q
30 = Q, so the social optimum is Q = 30 thousand tonnes.
The price consumers would pay at that output is P = 60 - 0.5(30) = 60 - 15 = $45 per tonne. The free market therefore over-produces by 50 - 30 = 20 thousand tonnes.
Step 3: the welfare loss from over-production
For every unit between the social optimum (Q = 30) and the free-market output (Q = 50), the marginal social cost exceeds the marginal social benefit, so each of those units subtracts from welfare. The total loss is the area of the triangle between the MSC curve and the demand curve over that range.
The base of the triangle is the range of over-production: 50 - 30 = 20 thousand tonnes. The height is the vertical gap between MSC and demand at the free-market output, Q = 50:
MSC at Q = 50: 30 + 0.5(50) = 30 + 25 = $55
Demand (MSB) at Q = 50: 60 - 0.5(50) = 60 - 25 = $35
Gap = 55 - 35 = $20 per tonne.
Welfare loss = 1/2 x base x height = 1/2 x 20 x 20 = $200 (in thousands, i.e. $200,000 per year). This is the money value of the harm the free market leaves uncorrected.
Step 4: the effect of a Pigouvian tax
A per-unit tax set equal to the marginal external cost internalises the externality. Here that tax is $20 per tonne. It raises the firms' effective marginal cost so that the after-tax supply curve becomes:
P = (10 + 0.5Q) + 20 = 30 + 0.5Q, which is exactly the MSC line.
The new market equilibrium is where this taxed supply meets demand, which is the social optimum already found: Q = 30 thousand tonnes at a consumer price of $45 per tonne. Producers receive $45 - $20 = $25 per tonne after handing the tax to the government.
Government revenue from the tax = tax per unit x quantity traded = $20 x 30 thousand = $600 (in thousands, i.e. $600,000 per year). The tax removes the $200,000 welfare loss and raises $600,000 that can be earmarked for river clean-up or compensation.
Recommendation
On these numbers the case for a $20 per tonne Pigouvian tax is strong, and I recommend it. It pushes output from 50 to the efficient 30 thousand tonnes, eliminates the $200,000 annual welfare loss, and turns the external harm into $600,000 of revenue that can fund the clean-up. Doing nothing leaves a real, recurring loss, so inaction is hard to defend.
Two cautions shape how I would implement it. First, the result depends on knowing the external cost is $20; if regulators over- or under-estimate it, the tax misses the optimum, so the rate should be reviewed as monitoring data improve. Second, if demand for the chemical is price inelastic, the tax will cut output less than this linear example suggests and will fall heavily on buyers, which can be regressive; the revenue should then be used to soften that burden. Where the external cost is hard to value or the total quantity of pollution matters most, a tradable-permit scheme that fixes the quantity may out-perform a tax. Subject to those checks, the tax is the efficient first choice here.