MAppII - Mathematical Appendix II II.1 Theorem of the...

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Unformatted text preview: Mathematical Appendix II II.1 Theorem of the Maximum There are two sets S IR n and T IR m . Further, there are a correspondence mapping S into the sets of subsets of T and a function f : S T IR. That is, ( x ) is a subset of T for every x S, and f ( x,t ) is a real number for every x S and t T. We are interested in the constrained maximization problem with f as the objective function and as the constraint. That is, given x S, max t f ( x,t ) (4) subject to t ( x ) . We denote by g ( x ) the maximized value of function f and by ( x ) the subset of vectors t in ( x ) on which the maximum value is attained. Formally, g ( x ) = max t ( x ) f ( x,t ) and ( x ) = { t ( x ) : f ( x,t ) = g ( x ) } . (5) Interpretation: Think about an economic agent whose environment is de- scribed by a vector x S. The agents set of actions is T, but when the environment is x, she is restricted to choose her action only from the sub- set ( x ) . Her utility of action t is f ( x,t ) , when the environment is x. Her objective is to choose an action in ( x ) to maximize her utility. 1 We shall assume that the set T is compact. Correspondence is said to be continuous if it is lower hemi-continuous and upper hemi-continuous. These are defined as follows: (LHC) is lower hemi-continuous at x if for every sequence { x n } in S converging to x and every t ( x ) , there exists a sequence { t n } in T such that t n ( x n ) and { t n } converges to t....
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MAppII - Mathematical Appendix II II.1 Theorem of the...

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