Berge's maximum theorem with two topologies on the action set

We give variants on Berge's Maximum Theorem in which the lower and the upper semicontinuities of the preference relation are assumed for two different topologies on the action set, i.e., the set of actions availabe a priori to the decision-maker (e.g. a household with its consumption set). Two new uses are pointed to. One result, stated here without a detailed proof, is the norm-to-weak* continuity of consumer demand as a function of prices (a property pointed to in existing literature but without proof or precise formulation). This improves significantly upon an earlier demand continuity result which, with the extremally strong 'finite' topology on the price space, is of limited interest other than as a vehicle for an equilibrium existence proof. With the norm topology on the price space, our demand continuity result acquires an independent significance - particularly for practical implementations of the equilibrium solution. The second application referred to establishes the continuity of the optimal plan as a function of the decision-maker's information (represented by a field of events in a probability spcace of states).

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http://eprints.lse.ac.uk/19358/1/Berge%27s_Maximum_Theorem_With_Two_Topologies_On_The_Action_Set.pdf