Demand correspondence ($x(p,w)$)

A demand correspondence assigns a nonempty set of chosen consumption bundles to each price and wealth pair. Every chosen bundle satisfies the budget constraint. A utility function rationalizes the demand correspondence when the chosen bundles are exactly its maximizers over the budget set.

Revealed preference infers rankings from choices: a chosen bundle is directly revealed at least as desirable as any bundle affordable in the same budget. Indirect comparisons use chains of such observations.

The weak indirect revealed preference is the reflexive closure of the transitive closure of a relation of direct revealed comparisons. Equivalently, it consists of finite chains, including chains of length zero. The diagonal convention is essential if the relation is to be an intersection of complete weak preference relations on all alternatives.

The congruence axiom says that an affordable alternative indirectly revealed at least as desirable as a chosen alternative must also be chosen. For a chosen bundle $x$ and an affordable unchosen bundle $y$, the axiom rules out $yRx$; because $xRy$, every strict-part-preserving completion ranks $x$ strictly above $y$.