Questions: Consumer Duality: Expenditure and Indirect Utility Functions

Question 1

A consumer maximizes utility with income m at prices p, reaching utility level ū. A second consumer minimizes expenditure to reach exactly ū at the same prices. Which of the following must be true?

• A. They choose different bundles because one is maximizing and the other is minimizing
• B. They choose the same bundle — utility maximization and expenditure minimization yield identical optimal choices at the margin
• C. The utility-maximizing consumer spends more, since the expenditure-minimizing problem imposes a tighter constraint
• D. The expenditure-minimizing consumer reaches a higher utility because she is using income more efficiently

Answer: (B)

This is the core duality result. At the optimum, both consumers satisfy the same first-order conditions: the marginal rate of substitution equals the price ratio. The utility-maximizing consumer spends exactly m to reach ū; the expenditure-minimizing consumer spends exactly m to reach ū — the same bundle, the same expenditure, the same utility. Duality is not two different solutions but one solution approached from two directions.

Question 2

Why is Hicksian (compensated) demand more useful than Marshallian demand for measuring the welfare cost of a price increase to a consumer?

• A. Hicksian demand reflects actual shopping behavior, so it better predicts real-world consumption changes
• B. Hicksian demand holds utility constant, isolating the pure substitution effect without mixing in the income effect
• C. Hicksian demand is easier to estimate from observed market prices and quantities
• D. Marshallian demand overestimates how much consumers substitute between goods when prices change

Answer: (B)

When a price rises, a consumer is affected in two ways: they substitute toward cheaper goods (substitution effect) and they become effectively poorer (income effect). Marshallian demand mixes both effects together, so a price change moves the consumer to a different utility level. Hicksian demand holds utility constant via a hypothetical income compensation, isolating only the substitution effect. For welfare analysis — measuring exactly how much a price change hurts — you need the pure substitution effect, which is what Hicksian demand provides.

Question 3

Shephard's lemma states that differentiating the expenditure function e(p, ū) with respect to the price of good i gives the Hicksian demand for good i.

• T. True
• F. False

Answer: True

Shephard's lemma is ∂e(p,ū)/∂pᵢ = hᵢ(p,ū), where hᵢ is Hicksian demand. Intuitively, if the price of good i rises by a small amount, the minimum expenditure needed to reach ū rises by approximately the quantity of good i being consumed — which is exactly the Hicksian demand. This is extremely useful because the expenditure function is often easier to derive analytically than solving directly for Hicksian demand from the expenditure-minimization first-order conditions.

Question 4

Marshallian demand holds utility constant when measuring a consumer's response to a price change.

• T. True
• F. False

Answer: False

This is the most common confusion between Hicksian and Marshallian demand. Marshallian demand holds *income* constant — it is what you observe when a price changes and income stays the same. As a result, utility changes along the Marshallian demand curve. Hicksian demand is the one that holds *utility* constant (with income adjusted to compensate). Confusing the two leads to incorrect welfare analysis: using Marshallian demand to measure welfare effects conflates how much a consumer substitutes with how much poorer they become.

Question 5

In your own words, what is the 'duality' in consumer duality, and why does it matter for welfare analysis?

Model answer: Consumer duality is the observation that utility maximization (choosing the best bundle given income) and expenditure minimization (spending the least to reach a target utility) are two descriptions of the same optimum. The 'duality' is that these two problems — one a maximum, one a minimum — have the same solution. It matters for welfare analysis because the expenditure function directly answers the policy question: 'How much extra income does a consumer need after a price increase to be just as well off as before?' That number is the compensating variation, and it can only be computed cleanly using Hicksian demand, which comes from the dual problem.

The duality framework separates the substitution and income effects of price changes with mathematical precision. Without it, measuring welfare effects of policy interventions — taxes, subsidies, trade liberalization — is imprecise because Marshallian demand confounds the two effects. Hicksian demand, derived from the dual expenditure problem, isolates the efficiency cost (deadweight loss) from the distributional cost (who bears the burden). Shephard's lemma and Roy's identity are the connecting identities that let you move between the primal and dual representations without re-solving optimization problems from scratch.