On the Characterization of Efficient Production Vectors

Abstract

International audienceIn this paper we study the efficient points of a closed production set with free disposal. We first provide a condition on the boundary of the production set, which is equivalent to the fact that all boundary points are efficient. When the production set is convex, we also give an alternative characterization of efficiency around a given production vector in terms of the profit maximization rule. In the non-convex case, this condition expressed with the marginal pricing rule is sufficient for efficiency. Then we study the Luenberger's shortage function. We first provide basic properties on it. Then, we prove that the above necessary condition at a production vector implies that the shortage function is locally Lipschitz continuous and the efficient points in a neighborhood are the zeros of it and conversely

Lipschitz properties of the scalarization function and applications

“…Moreover, when A ¼ ÀK and k 0 2 int K then (it is well known that) ' A is a continuous sublinear function, and so ' A is Lipschitz continuous. Recently in the case Y ¼ R m and for K ¼ R m þ Bonnisseau-Crettez [2] obtained the Lipschitz continuity of ' A around a point y 2 bd A when Àk 0 is in the interior of the Clarke tangent cone of A at y. The (global) Lipschitz continuity of ' A can be related to a result of Gorokhovik-Gorokhovik [9] established in normed vector spaces as we shall see in the sequel.…”

“…The cone K determines the order K on Y defined by y 1 K y 2 if y 2 À y 1 2 K. Throughout the article, we also assume that A satisfies the following condition (see also [2]):…”

“…However, local continuity properties were not studied in [8]. Very recently Bonnisseau and Crettez [2] obtained local Lipschitz properties for ' A,k 0 (called Luenberger shortage function in [2]) in a very special case. Of course, ' A,k 0 is a continuous sublinear functional if A is a proper closed convex cone and k 0 2 int A (cf [8,Corollary 2.3.5]) and so ' A,k 0 is Lipschitz continuous.…”

Properties of inefficiency indexes on 〈input, output〉 space

“…(In their papers on benefit functions, Luenberger (1992) and Briec and Garderes (2004) did prove some continuity results, but these results do not generalize to shortage functions.) Later, Chambers et al (1998) and Bonnisseau and Crettez (2007) were able to establish only upper semicontinuity of the shortage function; this is because they restricted the direction g only to be non-negative,whereas we restrict it to be strictly positive.…”

A special class of extended multicriteria location problems

“…[6]). We establish conditions under which the sets of efficient and weakly efficient solutions coincide -a property which play an important role in mathematical economics, as shown by Bonnisseau and Crettez [16], and Flores-Bazán, Laengle and Loyola [17]. In particular, we prove that this property holds for any bicriteria convex minimization problem provided that each criterion attains its minimum at a unique feasible point (which is the case in location problems).…”