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Integrability theory determines when demand functions can be recovered from underlying preferences through integration. The Slutsky matrix must be symmetric and negative semidefinite for consistency with utility-maximizing behavior. This connects observable demand to unobservable preferences through mathematical constraints.
From revealed preference axioms, you know how to test whether observed choices are consistent with utility maximization: if a consumer chose bundle A when B was affordable, they revealed a preference for A over B, and this pattern must be acyclical. Integrability asks the continuous version of the same question: given a smooth demand function x(p, m), can we find a utility function that generates it? This is the inverse problem — instead of deriving demand from preferences, we start with demand and work backward to preferences.
The answer hinges on the Slutsky matrix, which you encountered when studying compensated demand curves. The Slutsky matrix S has entries s_ij = ∂h_i/∂p_j, where h is Hicksian demand — the substitution effect of a price change holding utility constant. For a demand system to be rationalizable by some utility function, the Slutsky matrix must satisfy two conditions everywhere: it must be symmetric (s_ij = s_ji) and negative semidefinite (the substitution effect of any price change reduces compensated demand for that good). Symmetry means the cross-substitution effect of good j's price on good i's demand equals the reverse. Negative semidefiniteness means compensated demand curves slope downward.
Why symmetry? It comes from the mathematics of integration. If demand functions are generated by maximizing some utility function, the Hicksian demands are derivatives of the expenditure function: h_i = ∂e/∂p_i. The Slutsky matrix entries are then second derivatives: s_ij = ∂²e/∂p_i∂p_j. By Young's theorem (equality of cross-partials for smooth functions), these must be symmetric. This is exactly the condition needed to "integrate back" from demand to the expenditure function, and from there to the underlying utility. If the Slutsky matrix is asymmetric, no utility function can generate the observed demand — the demand system is fundamentally inconsistent with optimization.
The integrability theorem thus closes the circle between three ways of describing consumer behavior: preferences (ordinal utility), choice behavior (demand functions), and revealed preference (observed purchase data). If the Slutsky conditions hold, you can start from any one of these and recover the others. This matters practically because economists typically observe demand, not utility. Integrability tells you precisely when it is legitimate to estimate a demand system and interpret the results as reflecting coherent underlying preferences — and when the data reject the optimization hypothesis altogether.