Vladimir L. Levin

Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem

AbstractAbstract cyclical monotonicity is studied for a multivalued operator F : X → L, where L

Some applications of set-valued mappings in mathematical economics

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On preference relations that admit smooth utility functions

RX. A criterion for F to be L-cyclically monotone is obtained and connections with the notions of L-convex function and of its L-subdifferentials are established. Applications are given to the general Monge–Kantorovich problem with fixed marginals. In particular, we show that in some cases the optimal measure is unique and generated by a unique (up to the a.e. equivalence) optimal solution (measure preserving map) for the corresponding Monge problem.

A method in demand analysis connected with the Monge—Kantorovich problem

Abstract Reduced cost functions, introduced by the author in the context of the general mass transfer problem, have proved to be useful in some economic applications. In the present paper the properties of such functions and closely related sets Q 0 (c)= {u: X → R 1 : u(x) − u(y)⩽c(x,y)(x,yϵ X )} are examined in a more general setting than before. Three applications to mathematical economics are then considered, viz demand theory, rationalizability of action profiles in a principal-agent framework, and optimality of trajectories in dynamic optimization problems.

New axiomatic characterizations of utilitarianism

Abstract Given a separable metrizable space X and a metric d on it a characterization of preferences R:X →2 X that admit d -Lipschitz utility functions is presented. Also characterized are choice functions that can be rationalized by d -Lipschitz and by continuous utility functions. The asymptotic behavior of a dynamical system determined by R is another subject of study in the paper. The trajectories of such a system are sequences χ =( χ ( t )) ∞ t =0 with χ ( t )∈ R ( χ ( t -1)), t =1,2,...Properties are examined of a certain global attractor which, in the particular case of compact X and Hausdorff-continuous R , was introduced by Rubinov (1980) as an analogue of a turnpike in models of growth.

Abstract Convexity and the Monge-Kantorovich Duality

AbstractGiven a nonempty set X, we consider all cost functions c: X×X→ℝ1∪{+∞} and take the multifunction

Dual Representations of Convex Sets and Gâteaux Differentiability Spaces

Q_0 (c): = \{ u \in \mathbb{R}^X :u(x) - u(y) \leqslant c(x,y) for all x,y \in X\}