An abstract extremal principle with applications to welfare economics (Q1589956)

The author considers a model of welfare economics with a commodity space being a Banach space \(E\). The model involves \(n\) consumers with consumption sets \(C_i\subset E\), \(i=1,\dots,n\). Each consumer has a preference set \(P_i(x)\) that consists of elements in \(C_i\) preferred to \(x_i\) by this consumer at the consumption plan \(x=(x_i)\in C_1\times\cdots\times C_n\). The generalized preference relation is given by \(n\) multifunctions \(P_i\colon C_1\times\cdots\times C_n\to C_i\). The model also involves \(m\) firms, technological posibilities of which are represented by sets \(S_j\subset E\), \(j=1,\dots,m\). Given a net demand constraint set \(W\subset E\), \(x=(x_i)\in C_1\times\cdots\times C_n\) and \(y=(y_j)\in S_1\times\cdots\times S_m\), \((x,y)\) is feasible allocation if \(\sum_ix_i-\sum_jy_j\in W\). In this context the author considers three generalized notions of Pareto optimal allocations which reduce to conventional concepts of Pareto optimality for economic models with preference relations given by some preorder and/or utility functions. The main results are necessary conditions for all three notions of Pareto optimal allocations. The proof is based on reducing the three notions to locally extremal points of some systems of sets, and applying a new general extremal principle obtained in terms of abstract prenormal and normal structures in Banach spaces.