On the efficiency and optimality of random allocations (Q761335)

Allocations in a random economy described by an integrably bounded, closed and convex valued, measurable consumption multifunction and a monotone, continuous utility function are studied. The notions of efficiency and optimality of allocations are introduced and compared. Through the use of price systems belonging to \(L^{\infty}_{x^*}=(L^ 1_ x)^*\), a necessary and sufficient condition for optimality is obtained. Also for the case where the allocation belongs to \(L_ k^{\infty}\), it is shown that the efficiency prices can be chosen to be in \(L^ 1_{x^*}\), although \((L_ x^{\infty})^*\varsupsetneq L^ 1_{x^*}\). Finally, approximate optimality and efficiency are considered and also some stability results are proven when the consumption multifunction or the utility function vary in a certain sense. This work is based on the theory of normal integrands of \textit{R. T. Rockafellar} [see e.g. Lect. Notes Math. 543, 157-207 (1976; Zbl 0374.49001)].