Questions: Social Welfare Maximization and Optimal Taxation

Question 1

A government must raise revenue while minimizing deadweight loss. According to Ramsey's rule, which good should bear the highest tax rate?

• A. Luxury vacations, because high-income consumers can afford to pay more
• B. Gasoline, because transportation is a large share of household budgets
• C. Insulin, because its price elasticity of demand is very low
• D. Restaurant meals, because they are not necessities

Answer: (C)

Ramsey's rule says to tax goods in inverse proportion to their price elasticity: goods with the *lowest* elasticity should face the highest rates, because taxing them causes the smallest quantity reduction and therefore the smallest deadweight loss. Insulin demand is nearly perfectly inelastic (people need it regardless of price), so a tax raises revenue with minimal distortion. The other options reflect equity or political intuitions — taxing the rich or taxing luxuries — but these considerations are about distribution, not efficiency. This is exactly the equity-efficiency tension the topic explores.

Question 2

A social planner uses a Rawlsian welfare function. How does this change the optimal allocation compared to a utilitarian planner with identical preferences?

• A. The Rawlsian planner equalizes consumption across all individuals, just as the utilitarian planner does
• B. The Rawlsian planner focuses exclusively on improving the well-being of the worst-off individual, even at large aggregate cost
• C. The Rawlsian planner ignores redistribution entirely, favoring efficiency over equity
• D. The two planners produce identical allocations because both maximize total welfare

Answer: (B)

The Rawlsian (maximin) welfare function assigns all weight to the worst-off individual and none to others. Any transfer that raises the minimum utility level increases welfare, even if it decreases average utility substantially. Under utilitarianism with identical preferences, the planner equalizes consumption because the marginal utility of the last dollar is equal across people at that point — the utilitarian solution happens to be egalitarian, but for efficiency reasons, not because the planner cares specifically about the worst-off.

Question 3

Under a utilitarian social welfare function with identical preferences, the optimal allocation equates consumption across all individuals.

• T. True
• F. False

Answer: True

With identical, concave utility functions (exhibiting diminishing marginal utility), equalizing consumption maximizes the sum of utilities. Any transfer from a high-consumption individual to a low-consumption individual raises the recipient's utility by more than it reduces the donor's, because the marginal utility of consumption is higher for the poorer person. This continues until consumption is equalized. Note this result depends on identical preferences and concave utility — different preferences or less concave utility can yield unequal optimal distributions even under utilitarianism.

Question 4

Ramsey's principle of taxing goods with lower price elasticity conflicts with equity goals — but this tension disappears once we account for income effects.

• T. True
• F. False

Answer: False

The tension between Ramsey efficiency and equity is real and does not disappear with income effects. Goods with low price elasticity are often necessities (food, utilities, medicine) that are consumed disproportionately by low-income households. Taxing them heavily (as Ramsey recommends for efficiency) is regressive — it takes a larger share of income from the poor than from the rich. The income effect does not resolve this: it shifts the analysis slightly but does not eliminate the fundamental conflict. Different social welfare functions resolve the tension differently — a Rawlsian planner would weight equity so heavily that it substantially departs from Ramsey's efficiency prescription.

Question 5

Why does Ramsey's rule for optimal taxation create a conflict with equity, and how does the choice of social welfare function affect how this conflict is resolved?

Model answer: Ramsey's rule minimizes deadweight loss by placing higher taxes on goods with lower price elasticity. But goods with inelastic demand are typically necessities — food, medicine, utilities — that form a larger share of poor households' budgets. Taxing them most heavily is regressive. The choice of welfare function determines how to trade off this efficiency gain against the equity cost: a Rawlsian planner would accept larger deadweight losses to avoid taxing the poor, while a utilitarian planner might tolerate some regressivity to achieve greater aggregate efficiency, using the revenue for transfers that help the poor in other ways.

The core insight is that efficient tax design and equitable tax design point in opposite directions for necessities. Resolution requires a value judgment — the welfare function — not just economic analysis. Students often think Ramsey's rule is just 'bad policy' that ignores equity; the deeper point is that choosing a welfare function is a normative decision that economics can inform but not make.