Special Topics in Mathematical Economics and Optimization
In this chapter we have gathered certain specialized applications of multivalued analysis. So in section 1, we deal with the resource allocation problem from mathematical economics. This is a dynamic constrained optimization problem with infinite planning horizon. We present a relaxation method based on the so-called generalized plans. In contrast to the control-theoretic relaxation that we studied in chapter IV, in this model the utility has to be considered as part of the plan before the completion is performed. Otherwise, the utility of a generalized plan is not well-defined. We prove the existence of optimal generalized plans and characterize them. In section 2, we study the central question of including information as a parameter in the dynamic economic models and investigating how the various economic variables depend on this parameter. This requires the introduction of appropriate topologies in the space of information. Since information is modeled by sequences of sub-σ-fields of the original σ-field and the σ-fields can be identified with the corresponding conditional expectation operators, it is natural to look for convenient topologies in the realm of the operator topologies. Two such topologies are introduced and studied in detail. In particular, we examine how they can be used to obtain qualitative information in the context of economic models.
KeywordsSpecial Topic Convex Cone Choice Function Generalize Plan Stochastic Dominance
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