Transitive preferences in multi-member households

The classical result of Rose (Rev Econ Stud, 25:124–125, 1958) shows that, for single-member households, the Weak Axiom of Revealed Preferences (warp) and the Strong Axiom of Revealed Preferences (sarp) are equivalent when there are only two goods. Because sarp extends warp by requiring transitive preferences in addition, this means that transitivity of preferences need not be tested in the case of two goods. We consider the extension of this result towards \(L\)-member households, for which we introduce the concepts \(L\)-warp and \(L\)-sarp. For a general setting, we demonstrate that \(L\)-warp and \(L\)-sarp are not equivalent if there are at least \(L+1\) goods, which means that transitivity of individual preferences is testable. However, we can also show that \(L\)-warp and \(L\)-sarp do become equivalent for the restricted “labor supply” setting where we exclusively assign a single good to each different household member, i.e., \(L\) (out of \(L+1\)) goods are exclusive.