Questions: Duality: Expenditure and Indirect Utility

Question 1

A consumer with income $200 achieves utility level 30 at current prices. You then solve the expenditure minimization problem for the same consumer at the same prices, targeting utility level 30. What does the expenditure function e(p, 30) equal?

• A. Less than $200 — minimizing expenditure is more efficient than maximizing utility
• B. More than $200 — targeting a specific utility level is more expensive than a budget constraint allows
• C. Exactly $200 — the expenditure and indirect utility functions are mathematical inverses of each other
• D. It cannot be determined without knowing the specific form of the utility function

Answer: (C)

The expenditure and indirect utility functions are inverses: e(p, V(p, m)) = m and V(p, e(p, ū)) = ū. If $200 achieves utility 30 via utility maximization, then by the duality theorem the minimum expenditure needed to reach utility 30 at those same prices is exactly $200 — the same consumer, the same preferences, the same prices. The two optimization problems describe the same behavior from opposite directions, so they must agree on the expenditure at every optimum.

Question 2

A researcher wants Hicksian (compensated) demand functions that isolate the pure substitution effect. What is the most direct way to obtain them using duality theory?

• A. Solve the utility-maximization problem and apply the Slutsky equation to strip out the income effect
• B. Differentiate the expenditure function e(p, ū) with respect to each price — this is Shephard's lemma
• C. Differentiate the indirect utility function V(p, m) with respect to income m
• D. Solve the expenditure-minimization problem numerically for each price level

Answer: (B)

Shephard's lemma states that ∂e(p, ū)/∂pᵢ = hᵢ(p, ū), the Hicksian (compensated) demand for good i. This is the mathematical payoff of duality: instead of solving the minimization problem explicitly at each price, you differentiate the expenditure function once. The result automatically holds utility constant, eliminating the income effect. The Slutsky equation approach (option A) also works but is more roundabout.

Question 3

The duality between the expenditure function and indirect utility function means that a consumer solving the expenditure-minimization problem has different underlying preferences than one solving the utility-maximization problem.

• T. True
• F. False

Answer: False

Duality does not imply two different agents or preference structures. Both functions represent the same consumer with the same preferences — one viewed from the direction of maximizing utility given income, the other from the direction of minimizing expenditure given a utility target. The mathematical relationship e(p, V(p,m)) = m confirms they are inverses describing the same optimizing behavior. A core misconception to avoid is treating the dual problem as belonging to a 'different' consumer.

Question 4

If you know the expenditure function e(p, ū) for all prices and utility levels, you have in principle all the information needed to recover the indirect utility function V(p, m).

• T. True
• F. False

Answer: True

This follows directly from the duality relationship. Since e and V are inverses — e(p, V(p,m)) = m and V(p, e(p,ū)) = ū — knowing either one allows you to derive the other. They contain identical information about the consumer's preferences; duality is a statement that the primal and dual representations of preferences are mathematically equivalent, not just approximately equal.

Question 5

Why is duality theory useful in consumer analysis? Why not always solve the utility-maximization problem directly?

Model answer: Duality provides a mathematically cleaner route to Hicksian demands (via Shephard's lemma: differentiate the expenditure function) that directly isolates pure substitution effects without needing the Slutsky decomposition. Marshallian demands from utility maximization mix income and substitution effects, requiring extra steps to separate them. More broadly, duality reveals that the primal and dual approaches are two views of the same preferences — whichever is algebraically simpler can be used, and results derived from one automatically hold for the other.

The practical advantage is that the expenditure function is often easier to work with for welfare analysis (e.g., computing compensating variation and equivalent variation), while Marshallian demands are more natural for empirical estimation. Duality guarantees these approaches are consistent, unifying consumer theory rather than leaving two separate toolboxes.