Minimaximin results with applications to economic equilibrium (Q1057770)

We suppose that \(f: P\times C\to R\) and \(E: P\to 2^ C\) are given. This paper gives conditions under which we can assert that there exists \(q\in P\) such that, for all \(p\in P\), there exists \(z\in E_ q\) such that f(p,z)\(\leq 0\). This problem has its origin in the theory of Walrasian equilibrium (in which P is a set of price levels, C is a set of commodity bundles, f(p,c) is the price of commodity bundle c at price level p and \(E_ p\) is the excess demand at price level p). Our results represent a considerable generalization of recent results of \textit{C. D. Aliprantis} and \textit{D. J. Brown} [ibid. 11, 189-207 (1983; Zbl 0502.90006)] in that they apply to concave-convex functions and excess demand correspondences rather than bilinear functions and excess demand functions. We do not assume that P is compact, but assume instead that f and E satisfy a certain boundary condition. We consider separately the cases when the sets \(E_ p\) are compact and the case when the sets \(E_ p\) are not assumed to be compact.