A geometric approach to revealed preference via Hamiltonian cycles (Q2015040)

This paper deals with a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference implies the strong axiom of revealed preference. The main result of the present paper is the following. Theorem 1. Let \(T=\{x^{j}\}_{j\in{\mathcal J}}\) be a set of commodity bundles, with \({\mathcal J}\) a set of \(K\geq3\) distinct indices. The following conditions are equivalent: 1) There exists a set of price vectors \(\{p^{j}\}_{j\in{\mathcal J}}\) and a permutation \({\mathcal I}\) of \({\mathcal J}\) such that \(\{(x^{i},p^{i})\}_{i\in{\mathcal I}}\) forms a revealed preference cycle of irreducible length \(K\). 2) The convex monotonic hull of all elements of \(T\) admits a Hamiltonian cycle involving all elements of \(T\). Using this result the author proves that for the case of commodity space \(\mathbb R_{+}^{2}\) the weak axiom of revealed preference implies the strong axiom of revealed preference for any finite set \(S\) of data, and for the case of commodity space \(\mathbb R^{L}_{+}\) ,\(L>2\), there exist finite sets of observations which satisfy the weak axiom of revealed preference but not satisfy the strong axiom of revealed preference.