 Solution

Let $B(p,w)=\{z\in {\mathbb{R}}^L:p\cdot z\le w\}$ be the set allowed by the budget constraint. A demand correspondence is understood to choose only feasible bundles: $x(p,w)\subseteq B(p,w)$. Define the direct revealed preference relation byThe indirect revealed preference relation is its reflexive transitive closure. Thus $yRz$ precisely when there is a finite chain $y=y_0,y_1,\dots ,y_m=z$ whose successive comparisons satisfy $y_j{R}_D{y}_{j+1}$. Chains of length zero are included. This makes $R$ a preorder on all of ${\mathbb{R}}^L$, including alternatives that are never chosen. The associated strict preference isIf a convention defines indirect revelation using only positive-length chains, one must add the diagonal before the intersection assertion in part (c) can hold: every complete weak preference relation contains the diagonal.