Demand Systems and Integrability Conditions

Core Idea

Not all demand functions can come from maximizing a utility function; they must satisfy integrability conditions (Slutsky symmetry, negative semi-definiteness). These conditions ensure the demand system is consistent with optimization. Testing whether observed demand satisfies integrability reveals whether consumer behavior is rational without knowing preferences.

Explainer

From your work on duality, you know that a utility-maximizing consumer generates demand through two equivalent routes: direct utility maximization (Marshallian demands) and expenditure minimization (Hicksian demands), linked by the Slutsky equation. The integrability question inverts this: given an observed demand function x(p, w), does there exist a utility function that rationalizes it? The answer is: not always. Integrability conditions are the set of restrictions that a demand function must satisfy to be consistent with utility maximization.

The key object is the Slutsky matrix — the matrix of compensated price effects, with entry (i, j) equal to ∂x_i/∂p_j + x_j·(∂x_i/∂w). For demand generated by utility maximization, this matrix must satisfy two conditions. First, Slutsky symmetry: the (i, j) entry must equal the (j, i) entry, meaning the compensated cross-price effect of good j on good i equals the compensated cross-price effect of good i on good j. Second, negative semi-definiteness: the matrix can have no positive eigenvalues, meaning compensated own-price effects are non-positive. Both conditions follow directly from the properties of the expenditure function you derived in your duality study.

These conditions have concrete economic content. Symmetry says that how coffee consumption responds to a compensated increase in tea prices must equal how tea consumption responds to a compensated increase in coffee prices. This is an observable, testable implication of rationality — not an assumption about any particular preference shape, but a structural requirement of optimization itself. Negative semi-definiteness says that compensated demand slopes downward: hold utility constant, raise a price, and the consumer buys no more of that good.

The power of integrability conditions is that they let you test rationality from demand data without ever observing preferences directly. If you estimate a demand system from household expenditure surveys and find that the Slutsky matrix is asymmetric or has a positive eigenvalue, you have evidence against utility-maximizing behavior — without specifying what utility function the consumer "should" have. This is the bridge between the axiomatic revealed preference framework you studied and empirical demand analysis: integrability conditions are how you check whether a system of observed demand functions is consistent with a coherent underlying optimization problem.