Marrow Vale's Sugar Tax: Pigouvian Pricing and Fairness
Marrow Vale, a city of 1.4 million, is designing a tax on sugar-sweetened drinks. Annual demand in million litres is estimated as Qd = 250 − 50P, where P is the price per litre in dollars. Domestic producers' marginal private cost corresponds to Qs = 50P + 10. Public-health economists estimate an external cost of $0.60 per litre from obesity-related healthcare and lost productivity, and there is no external benefit. Household survey data show that the bottom income quintile has a price elasticity of demand for these drinks of −0.3, while the top quintile's elasticity is −1.4. The city's health budget is rising 7% a year and the council has promised not to raise property taxes. Retailers warn that a large tax will push sales into neighbouring municipalities and that small corner shops, which earn 12% of their revenue from these drinks, will lose out. The council must choose between a specific tax of $0.60 per litre, a subsidy for healthier alternatives, and a voluntary reformulation agreement with manufacturers.
1. Calculate the free-market equilibrium price and quantity and the socially optimal quantity. Calculate the deadweight loss at the free-market equilibrium and explain its source without drawing a diagram.
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Free market. Set Qd = Qs: 250 − 50P = 50P + 10, so 240 = 100P and P = $2.40, giving Q = 130 million litres per year.
Social optimum. Marginal social cost equals marginal private cost plus the external cost. Inverting supply: P = Q/50 − 0.2, so MSC = Q/50 − 0.2 + 0.6 = Q/50 + 0.4. Inverting demand: P = 5 − Q/50. Setting MSB = MSC: 5 − Q/50 = Q/50 + 0.4, so 4.6 = 2Q/50 and Q = 115 million litres. Consumers value the 115th litre at 5 − 2.3 = $2.70.
Overproduction and deadweight loss. At Q = 130, MSC = 130/50 + 0.4 = $3.00 while buyers value the last litre at only $2.40; the gap is the $0.60 marginal external cost. Every litre between 115 and 130 costs society more than it is worth to consumers, yet it is produced because buyers ignore the health cost borne by others. The loss is the triangle 0.5 × (130 − 115) × 0.60 = $4.5 million per year.
Explanation in words. Consumption generates a negative externality, so marginal private benefit exceeds marginal social benefit and the free market quantity is too large. The deadweight loss measures the surplus destroyed by those excess units, not the external cost itself (which is 0.60 × 130 = $78 million of external cost at the free-market quantity). The estimate is static and ignores any benefit of reduced illness to non-consumers and any effect of advertising.
2. Calculate the effect of a $0.60 per litre specific tax. Explain who bears the burden and comment on the tax's fairness using the elasticity data.
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New equilibrium. The tax shifts supply up by $0.60, so producers receive P − 0.60 and Qs = 50(P − 0.60) + 10 = 50P − 20. Setting 250 − 50P = 50P − 20 gives 270 = 100P, so the consumer price is $2.70 and Q = 115 million litres, exactly the social optimum. Producers receive 2.70 − 0.60 = $2.10. Tax revenue = 0.60 × 115 = $69 million per year.
Incidence. Consumers pay $0.30 more than before ($2.40 to $2.70) and producers receive $0.30 less, so the burden is split equally. This matches the elasticities: at the original equilibrium |PED| = 50 × 2.40/130 ≈ 0.92 and PES ≈ 0.92, and the consumer share is PES ÷ (PES + |PED|) = 0.5. Quantity falls 11.5%, tax revenue is large, and the deadweight loss is eliminated because the tax equals the marginal external cost.
Fairness. The single elasticities hide a distributional problem. Low-income consumers (|PED| = 0.3) barely change what they buy, so they pay most of the price rise as a share of income: the tax is regressive. High-income consumers (|PED| = 1.4) switch to substitutes and largely avoid it. Set against this, diet-related disease is concentrated in low-income groups, so the health gain is also progressive, and the revenue could be recycled to them. There is also horizontal inequity: people who buy the drinks rarely still face no cost, while some heavy consumers with medical conditions pay a lot. Two further cautions: a city-level tax is easy to avoid by crossing into neighbouring municipalities, so the effective local elasticity is probably far higher than −0.3, and small shops dependent on these drinks lose turnover. The overall judgement is that the tax is efficient but needs revenue recycling to be fair.
3. Evaluate the three options - the $0.60 tax, a subsidy for healthier alternatives, and a voluntary reformulation agreement - and recommend one.
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Criteria. Judge each option on effectiveness in reducing the external cost, efficiency (surplus and administration), equity, fiscal cost and enforceability.
Tax. It directly closes the gap between marginal private and social cost, moves quantity to the social optimum of 115 million litres, and raises $69 million that can be recycled. It works through a price signal rather than a bureaucracy and needs little information once the external cost is estimated. Weaknesses: regressive incidence given the −0.3 elasticity of low-income buyers, the external cost estimate is uncertain, cross-border shopping erodes effectiveness for a single city, and small retailers lose revenue. If the tax exceeds the true marginal external cost, it creates a new deadweight loss.
Subsidy for healthier alternatives. It lowers the price of a substitute and can be designed to help low-income households, treating them as the main beneficiaries rather than the main payers. But it costs the budget rather than raising revenue, the benefit is partly captured by producers unless supply is elastic, 'healthy' must be defined and monitored, and it reduces the externality only indirectly.
Voluntary reformulation. It can cut sugar content without any price rise, is cheap and quick, and industry may cooperate to avoid regulation. But it lacks enforcement, invites free-riding between firms, has no revenue, and targets are hard to verify; experience suggests agreements of this type deliver less than promised where competition is strong.
Recommendation. Adopt the $0.60 tax as the core instrument because it is the only option that reliably corrects the externality and monetises it, but pair it with revenue recycling: fund the healthier-alternative subsidy and targeted support for low-income households from the $69 million, and monitor cross-border sales. Treat reformulation as a complementary measure with binding, published targets and penalties. Justify the ranking explicitly: a tax alone would be efficient but unfair in incidence given the elasticity differences; a subsidy alone would be fairer but fiscally costly and weaker at correcting the externality; voluntary action alone is unlikely to be enforced. A good answer also flags that if cross-border leakage proves severe, the policy should be set at regional or national level rather than city level.