A property of allocation rules (in problems involving fair division of infinitely divisible goods amongst a finite number of agents) that has received considerable attention in recent literature is consistency. The property may be described as follows: apply a allocation rule to a problem within a chosen class of problems. consistency would require that the restriction of any allocation chosen by the rule for that problem to any subgroup of agents is what this solution would recommend for the “reduced problem” obtained by imagining the departure of the members of the complementary group with what they receive, and re-evaluating the situation from the view point of those who remain. In Lahiri [1997] can be found axiomatic characterization of the equal income market equilibrium choice correspondence in economies characterized by convex preferences, and using the properties of consistency and converse consistency. The earliest known work in a similar direction is the one by Thomson [1988] followed by Thomson [1994] where consistency plays a fundamental role. In the Lahiri [1997] paper, the second fundamental theorem of welfare economics plays a crucial role. Young [1993] provides a generalization of a Thomson [1988] result, where a characterization of sub-correspondences of the equal income competitive equilibrium allocations are available (for convex preferences) using consistency and a property called replication invariance. Simply put, replication invariance says that if we replicate an allocation problem, the solution outcomes also get replicated. In this paper, we extend the Young [1993] result to economies where preferences may be non-convex. We invoke a property called Sigma optimality which Svennson [1994] uses rather persuasively to establish the existence of fair allocations in non-convex economies. The corresponding welfare theorem in terms of optimality is used to characterize the equal income market equilibrium choice correspondence in terms of consistency replication invariance, and a property called quasi-local independence. The last property is implied by local independence – a property used repeatedly in Lahiri [1997]. For a general survey of the literature concerning problems of fair division, the reader should refer to Thomson and Varian [1985].