Budget-constrained pareto efficient allocations: A dynamic story
It has been established in the context of pure exchange economies that, given a fixed price system and a given income distribution, there exists at least one allocation which is going to be Pareto efficient and still satisfy every consumer’s budget equation with respect to the given price system and the income assigned to him. Existence, structural stability, and uniqueness of B.C.P.E. allocations have been dealt with in [ 11 under the assumption of differentiability of preferences; the existence result has been extended so as to cover non-differentiable preferences in 121. The theory developed in these papers being basically static, the purpose of this note is therefore to add a dynamic touch to the theory of B.C.P.E. allocations. We use the same definitions, assumptions, and notation as in [Il. If 2 = (p, WI, w, ,*..r wm) denotes a price-income vector with p.r = w,+W,+**~+W,, where r is the vector of total resources, we denote by A(i) the space of budget-constrained allocations associated with 2. The space A(Z) is a linear manifold having dimension (f 1) (m 1). We define X as the set of allocations x = (x,, x2 ,..., x,J with C xi = r. In this note, we shall investigate processes that lead to B.C.P.E. allocations in a sense to be made more precise later on.