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Higher-order conditions for local equilibria in a discontinuous Gale economic model

Michalak A., Studniarski M.

Control and Cybernetics

2017

Vol. 46, no. 1

37--47

The paper introduces the concept of a strict local equilibrium of order k in the Gale economic model. We obtain higher-order necessary and suﬃcient conditions for such equilibria without assuming continuity of the utility functions. These conditions are formulated in terms of generalized lower and upper directional derivatives, introduced by Studniarski (1986). A stability theorem for strict local equilibria of order k is also included.

Necessary and sufficient conditions for a Pareto optimal allocation in a discontinuous Gale economic model

Michalak A., Studniarski M.

Opuscula Mathematica

2014

Vol. 34, no. 4

827--835

In this paper we examine the concept of Pareto optimality in a simplified Gale economic model without assuming continuity of the utility functions. We apply some existing results on higher-order optimality conditions to get necessary and sufficient conditions for a locally Pareto optimal allocation.

Generalized directional derivatives for locally Lipschitz functions which satisfy Leibniz rule

Grzybowski J., Pallaschke D., Urbański R.

Control and Cybernetics

2007

Vol. 36, no 4

911-924

In this paper a concept of a generalized directional derivative, which satisfies Leibniz rule is proposed for locally Lipschitz functions, defined on an open subset of a Banach space. Although Leibniz rule is of less importance for a subdifferential calculus, it is of course of some theoretical interest to know about the existence of generalized directional derivatives which satisfy Leibniz rule. The proposed concept of generalized directional derivatives is adopted from the work of D. R. Sherbert (1964) who determined all point derivations for the Banach algebra of Lipschitz functions over a complete metric space.