Existence of budget constrained pareto efficient allocations
The concept of a budget constrained Pareto efficient (BCPE) allocation was introduced by Balasko (11, who gave an existence proof in the differentiable case using a duality approach. Intuitively, a BCPE allocation (X ,,..., xm) is Pareto efficient, and, in addition, the bundle of consumer i satisfies a budget constraint qxt = qwi, where q E IR: is a given, fixed price vector and wi the initial endowment of consumer i. In the present note we give a direct proof of the existence of BCPE allocations under rather weak conditions: thus preferences are not necessarily total or transitive. Only convexity of the set of preferred bundles together with some monotonicity and continuity conditions are assumed, and we use the approach to existence proofs initiated by Gale and Mas-Cole11 [3] and Shafer and Sonnenschein [4]. An application of the model to the problem of fair division of a random harvest introduced by Gale [2] is indicated. Let B = ((Xi, Pi)?:, , w) be an exchange economy, where for i= l,...,m: