Questions: Duality in Consumer Theory: Utility and Expenditure

Question 1

An economist wants to find how much of good i a consumer purchases to achieve utility ū as cheaply as possible, as prices change. Which function should they differentiate, and with respect to what?

• A. Differentiate the indirect utility function V(p, m) with respect to m
• B. Differentiate the expenditure function e(p, ū) with respect to p_i — this gives Hicksian demand for good i by Shephard's lemma
• C. Differentiate the Marshallian demand x(p, m) with respect to p_i
• D. Differentiate the utility function u(x) with respect to x_i

Answer: (B)

Shephard's lemma states that ∂e(p, ū)/∂p_i = h_i(p, ū), the Hicksian (compensated) demand for good i. This is the remarkable practical power of duality: the entire compensated demand system is encoded in the partial derivatives of a single scalar function. Marshallian demand (option C) mixes income and substitution effects; Hicksian demand isolates substitution effects by holding utility constant, which is what the expenditure minimization problem achieves.

Question 2

A government wants to estimate the monetary cost to a consumer of a 20% increase in the price of heating oil, holding the consumer's welfare constant. Which approach, grounded in duality theory, gives the theoretically cleanest answer?

• A. Compare the consumer's Marshallian demand for heating oil before and after the price change
• B. Compute the change in the expenditure function e(p, ū) at the new versus old prices — this is the compensating variation
• C. Subtract the new price from the old price and multiply by quantity demanded
• D. Use the indirect utility function to compute the change in utility, then convert to dollars using the marginal utility of income

Answer: (B)

Compensating variation is defined as the change in minimum expenditure needed to maintain the original utility level: e(p_new, ū) − e(p_old, ū). The expenditure function is the right tool because it holds utility constant (the 'dual' perspective) while varying prices. Marshallian demand (option A) mixes income and substitution effects, making welfare comparisons theoretically muddier. Duality makes the expenditure function the natural instrument for welfare analysis.

Question 3

At the consumer's optimum, the utility-maximizing consumer (primal problem) and the expenditure-minimizing consumer (dual problem) are solving equivalent problems.

• T. True
• F. False

Answer: True

This is the core claim of duality in consumer theory. At the optimum, both problems reach the same consumption bundle. The consumer who maximizes utility on a fixed budget and the consumer who minimizes cost to reach a fixed utility level are describing the same underlying preference-constrained tradeoff from opposite directions. The primal gives Marshallian demand; the dual gives Hicksian demand — but both describe the same indifference curve tangency condition.

Question 4

Marshallian demand is preferred over Hicksian demand for welfare analysis because it holds utility constant while prices vary.

• T. True
• F. False

Answer: False

This reverses the description. Hicksian (compensated) demand holds utility constant — it traces how the consumer substitutes between goods as prices change while staying on the same indifference curve. Marshallian demand holds income constant, which means utility changes as prices change. For welfare analysis, holding utility constant is exactly what is needed to measure the cost of a price change in money terms (compensating or equivalent variation). This is precisely why the expenditure function and Hicksian demand, not Marshallian demand, are the tools of welfare analysis.

Question 5

What is duality in consumer theory, and why does it matter for welfare analysis?

Model answer: Duality is the formal result that the utility function and the expenditure function are equivalent representations of the same underlying preference structure — one expressed as a maximum utility problem, the other as a minimum cost problem. They contain the same information in different algebraic forms. For welfare analysis, duality matters because the expenditure function directly answers 'how much money would make this consumer as well off as before?' — a welfare measure. Since utility and expenditure are dual to each other, economists can freely choose whichever representation is computationally convenient.

The practical implication is Shephard's lemma and Roy's identity: entire demand systems can be recovered by differentiating a single function. Welfare changes are computed as differences in the expenditure function rather than through complex integration of demand systems. Duality converts what would otherwise be a hard empirical problem (estimating preferences) into an exercise in calculus on well-behaved functions.