Irreducible economies and strongly connected graphs

In this paper we apply graph theoretic techniques to provide results for weakening the interior endowment assumption of Arrow and Debreu [Arrow, K.J., Debreu, G., 1954. Existence of an equilibrium for a competitive economy, Econometrica 22, 265-290]. We develop two conditions - C-irreducibility and C′-irreducibility - which are sufficient for the existence of competitive equilibria in economies with weakly monotone preferences and boundary endowments. We show that C-irreducibility is different from, and C′-irreducibility weaker than, McKenzie's irreducibility [McKenzie, L.W., 1959. On the existence of general equilibrium for a competitive market, Econometrica 27, 54-71; McKenzie, L.W., 1961. On the existence of general equilibrium: some corrections, Econometrica 29, 247-248]. We also provide conditions under which McKenzie's irreducibility and Arrow and Hahn's [Arrow, K.J., Hahn, F., 1971. General Competitive Analysis. Holden-Day, San Francisco] resource relatedness are equivalent. This answers an open question in the literature.