In recent times, beginning with the work of Thomson [1988], attention in social choice theory has focused on problems of fair division. The problem is one of dividing a social endowment among a group of agents keeping in mind the issues of equity and efficiency. There is a sizeable literature which has dealt with the problem of axiomatically characterizing the equal income market equilibrium solution. Notable among them are the works of Thomson [1988], Lahiri [1997, forthcoming]. In Thomson and Varian [1985], there is the concept of an income fair allocation. Basically, these are allocations which cost the same for all agents. However, the pirce vector at which the allocation is evaluated, is in the existing literature, endogenously determined by market forces, whence it coincides with equal income market equilibrium allocations. The income fairness concept derives appeal as an independent entity, only if the price at which the allocation is evaluated is specified exogenously. Along with efficiency, it then coincides with what Balasko [1979], calls budget constrained Pareto efficient allocations. To be relevant to the literature on fair division, it should be required that the monetary worth of the allocation at exogenously specified prices, be the same for all agents. In this paper, we axiomatically characterize the social choice correspondence, which picks for each economy, the set of equal income budget constrained Pareto efficient allocations. We are able to characterize this social choice correspondence uniquely, with the help of the following assumptions: Consistency, Equal Budget Property, Pareto efficiency for two agent problems and Local Independence. The most extensive use of Consistency is due to Thomson, as surveyed in Thomson [1990]; the equalbudget property is due to Varian [1976], Pareto efficiency for two agent problems is due to van der Nouweland, Peleg and Tijs [1996], Local Independence is due to Nagahisa [1991]. In a final section, we replace Consistency by Converse Consistency (Thomson [1990]) and Pareto Efficiency for two agent problems by Binary Efficiency (see Lahiri [1997] to obtain yet another axiomatic characterization of the same social choice correspondence.