Questions: Demand Systems and Integrability Conditions

Question 1

An economist estimates a demand system from household survey data and finds that the compensated cross-price effect of good A on good B differs from the compensated cross-price effect of good B on good A. What does this imply?

• A. The consumers have unusual preferences and need a more flexible utility function
• B. The demand system fails Slutsky symmetry and cannot be rationalized by any utility function
• C. The estimation is merely imprecise and the difference is likely sampling error
• D. The consumers violate budget balance but could still be utility-maximizing

Answer: (B)

Slutsky symmetry — that the (i,j) and (j,i) entries of the Slutsky matrix are equal — is a necessary condition for a demand system to be consistent with utility maximization. If this condition fails, no utility function exists that could generate the observed demands, regardless of how flexible that function is. The observed asymmetry is direct evidence against rationality as utility maximization, not a sign of unusual preferences or estimation noise.

Question 2

Which pair of conditions must the Slutsky matrix satisfy for a demand system to be integrable?

• A. Symmetry and positive definiteness
• B. Symmetry and negative semi-definiteness
• C. Homogeneity of degree zero and negative semi-definiteness
• D. Symmetry alone — semi-definiteness follows automatically from symmetry

Answer: (B)

Integrability requires both Slutsky symmetry (the matrix equals its transpose) and negative semi-definiteness (no positive eigenvalues, implying compensated own-price effects are non-positive). Symmetry alone is not sufficient — a symmetric matrix can have positive eigenvalues. Both conditions are independently necessary and jointly sufficient, flowing from the properties of the expenditure function derived under utility minimization.

Question 3

Slutsky symmetry is an observable implication of rationality that can be tested using demand data without ever directly observing a consumer's preferences.

• T. True
• F. False

Answer: True

This is precisely the power of integrability conditions. Slutsky symmetry — that compensated cross-price effects are equal across goods — is a restriction on the structure of observable demand, not on the shape of an unobservable utility function. Economists can estimate the Slutsky matrix from expenditure survey data and test symmetry statistically without specifying or observing preferences. It translates the abstract axiom of rationality into a falsifiable empirical claim.

Question 4

If a demand system satisfies Slutsky symmetry, it is expected to be consistent with utility maximization.

• T. True
• F. False

Answer: False

Symmetry is necessary but not sufficient. The Slutsky matrix must also be negative semi-definite — meaning compensated own-price effects are non-positive. A symmetric Slutsky matrix could still have a positive eigenvalue, which would violate the requirement that holding utility constant, a price increase cannot cause a consumer to buy more of that good. Both conditions together are required to guarantee integrability.

Question 5

Why do integrability conditions matter for empirical demand analysis, and what failure do they detect?

Model answer: Integrability conditions (Slutsky symmetry and negative semi-definiteness) determine whether an estimated demand system is consistent with utility-maximizing behavior. If an estimated Slutsky matrix is asymmetric or has a positive eigenvalue, the data reject the rationality hypothesis — no coherent optimization problem could generate those demands. This lets researchers test rationality from observable demand data without assuming a specific utility function.

The key insight is that integrability conditions bridge theory and data: they translate the abstract requirement of 'rational optimization' into testable restrictions on the structure of demand. Without these conditions, any demand function could claim to be rational. With them, rationality makes falsifiable predictions about the relationship between compensated cross-price effects — predictions that can be checked with household expenditure surveys.