A consumer faces prices p and has income m. She solves her utility maximization problem and finds indirect utility v(p, m) = u*. A researcher then minimizes the expenditure needed to achieve u* at the same prices. What will the researcher find?
AThe expenditure will be less than m, because expenditure minimization is more efficient than utility maximization
BThe expenditure will be exactly m, by the duality identity e(p, v(p,m)) = m
CThe expenditure will be greater than m, because the researcher targets a utility level rather than a budget
DThe expenditure cannot be determined without knowing the specific utility function
This is the core duality identity: e(p, v(p, m)) = m. The indirect utility v(p, m) is the maximum utility achievable with budget m at prices p. Asking 'what is the cheapest way to achieve that exact utility level at the same prices?' must yield m — because that is exactly what the consumer is already spending to achieve it optimally. The two problems are mirrors of each other; solving one and plugging the answer into the other returns you to the starting point.
Question 2 Multiple Choice
Why do economists use Hicksian (compensated) demand rather than Marshallian (uncompensated) demand for welfare analysis of price changes?
ABecause Hicksian demand is easier to observe in market data than Marshallian demand
BBecause Hicksian demand holds income constant while Marshallian demand holds utility constant
CBecause Hicksian demand isolates the pure substitution effect by holding utility constant, making it the right tool for measuring welfare changes
DBecause Marshallian demand violates the axioms of revealed preference theory
When a price changes, a consumer's real welfare changes — they can afford a different utility level. Marshallian demand, which holds income constant, mixes together the substitution effect (how the consumer substitutes away from the more expensive good) with an income effect (the change in real purchasing power). For welfare analysis, we want to isolate how much money the consumer would need to compensate for the price change — which requires holding utility constant. Hicksian demand does exactly this, making it the right tool for welfare analysis, even though it is not directly observable in market data.
Question 3 True / False
Duality in consumer theory means that the utility maximization problem and the expenditure minimization problem are two different theories of consumer behavior that yield different predictions.
TTrue
FFalse
Answer: False
Duality means they are the same theory viewed from two different angles — equivalent windows into the same underlying preferences. They generate the same demand behavior, connected by exact identities: h(p, u) = x(p, e(p, u)) and Marshallian demand via Roy's identity equals Hicksian demand via Shephard's lemma evaluated at the same optimum. A researcher using either formulation will reach the same empirical predictions; the choice is analytical convenience, not theoretical commitment.
Question 4 True / False
Shephard's lemma states that differentiating the expenditure function with respect to a price gives the corresponding Hicksian demand: ∂e/∂pᵢ = hᵢ(p, u). This means you can recover all demand information from the expenditure function alone, without solving the minimization problem directly.
TTrue
FFalse
Answer: True
This is one of the most powerful practical consequences of duality. If you know the expenditure function e(p, u) — perhaps estimated econometrically from observed data — you can recover the entire Hicksian demand system by differentiation. Similarly, Roy's identity extracts Marshallian demands from the indirect utility function. This means a researcher never needs to solve both problems: either form, once known, contains all the demand information needed for the other. Duality transforms what could be two separate estimation problems into one.
Question 5 Short Answer
What is the economic meaning of the duality identity v(p, e(p, u)) = u, and why does it hold exactly rather than approximately?
Think about your answer, then reveal below.
Model answer: The identity says: if you compute the minimum expenditure needed to reach utility u at prices p, and then ask what maximum utility you can achieve by spending that amount at those same prices, you get back exactly u. It holds exactly because both problems share the same optimum — at the optimal bundle, the utility-maximizing consumer and the expenditure-minimizing consumer make the same choice. There is no approximation because duality is not a simplification of the theory; it is a structural property of the optimization problems themselves. The optimum is unique (under standard regularity conditions), so both problems converge on the same bundle and the identities hold with equality.
The exactness of duality identities is what makes them so powerful analytically. They are not convenient approximations like linear demand approximations or log-linearizations — they are precise algebraic relationships between objects defined by the same underlying preference structure. This means any insight, comparative static, or welfare measure derived from one formulation can be exactly translated into the other.