An economist estimates demand and finds the cross-substitution effects s₁₂ = 0.3 and s₂₁ = 0.5 (where sᵢⱼ = ∂hᵢ/∂pⱼ, the compensated cross-price effect). What does this asymmetry imply?
AThe goods are substitutes, since both cross-effects are positive
BThe demand system is inconsistent with utility maximization — an asymmetric Slutsky matrix means no utility function can generate this demand
CIncome effects are dominating substitution effects, explaining the asymmetry
DThe result is plausible and common in empirical work; the Slutsky matrix need not be symmetric
Slutsky symmetry (sᵢⱼ = sⱼᵢ) is a necessary condition for integrability — it follows from Young's theorem applied to the expenditure function. If sᵢⱼ ≠ sⱼᵢ, no expenditure function exists whose cross-partials match the observed demand, which means no utility function can rationalize the behavior. An asymmetric Slutsky matrix is a rejection of the utility-maximization hypothesis. Income effects explain asymmetry in uncompensated (Marshallian) demands, but not in compensated (Hicksian) demands, which are what Slutsky entries measure.
Question 2 Multiple Choice
Why must the Slutsky matrix be symmetric for a demand system to be rationalizable by utility maximization?
ASymmetry is an empirical regularity imposed by revealed preference axioms, not derived from mathematics
BHicksian demands are derivatives of the expenditure function, and Young's theorem requires mixed partial derivatives to be equal: ∂²e/∂pᵢ∂pⱼ = ∂²e/∂pⱼ∂pᵢ
CThe law of demand requires that all substitution effects be equal across goods
DConsumers must treat symmetric pairs of goods identically for preferences to be well-defined
If a utility function exists, the expenditure function e(p, u) can be derived from it. Hicksian demands are hᵢ = ∂e/∂pᵢ, so Slutsky entries are sᵢⱼ = ∂²e/∂pᵢ∂pⱼ. For any smooth function, Young's theorem guarantees these mixed partials are equal. If the observed Slutsky matrix is asymmetric, you cannot construct such an expenditure function — the 'integration back' from demand to preferences fails. This is why symmetry is both necessary and sufficient (with negative semidefiniteness) for rationalizability.
Question 3 True / False
If the Slutsky conditions (symmetry and negative semidefiniteness) hold for an observed demand system, it is possible to recover a utility function consistent with that demand.
TTrue
FFalse
Answer: True
This is the content of the integrability theorem: the Slutsky conditions are necessary AND sufficient for rationalizability. If both hold, one can integrate the expenditure function from the Hicksian demands, then invert to recover the utility function. The demand system, the expenditure function, and the utility function are three equivalent representations of the same consumer behavior, and the Slutsky conditions are the key that unlocks movement between them.
Question 4 True / False
A negative semidefinite Slutsky matrix means most cross-substitution effects are negative — most goods are complements.
TTrue
FFalse
Answer: False
Negative semidefiniteness constrains the OWN-price substitution effects: it requires that compensated own-price effects sᵢᵢ ≤ 0 (compensated demand curves slope downward). Cross effects sᵢⱼ can be positive (substitutes) or negative (complements) and are unconstrained by negative semidefiniteness. NSD is a matrix condition on quadratic forms: for any vector v, v'Sv ≤ 0. This is satisfied even when many individual cross-entries are positive.
Question 5 Short Answer
What is the integrability problem in consumer theory, and why does Slutsky symmetry determine whether preferences can be recovered from observed demand data?
Think about your answer, then reveal below.
Model answer: The integrability problem asks: given an observed demand function x(p, m), can we find a utility function that generates it? Starting from demand, we must 'integrate back' to recover the expenditure function and then the utility function. This is only possible if the Slutsky matrix — whose entries are second derivatives of the expenditure function — is symmetric. Asymmetry means no smooth expenditure function exists, so no utility function can rationalize the observed demand. Symmetry (plus negative semidefiniteness) is both necessary and sufficient for the integration to succeed.
Integrability connects the three representations of consumer behavior: utility functions, demand functions, and revealed choice data. The Slutsky conditions are the mathematical bridge. Without symmetry, demand data are inconsistent with optimization — the consumer cannot be modeled as maximizing any stable preference ordering. This makes Slutsky symmetry not just a theoretical nicety but an empirical test of the rationality hypothesis itself.