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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 ϕ mapping S into the sets of subsets of T and a function f : S × T → IR. That is, ϕ ( 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 ϕ as the constraint. That is, given x ∈ S, max t f ( x,t ) (4) subject to t ∈ ϕ ( x ) . We denote by g ( x ) the maximized value of function f and by μ ( x ) the subset of vectors t in ϕ ( x ) on which the maximum value is attained. Formally, g ( x ) = max t ∈ ϕ ( x ) f ( x,t ) and μ ( x ) = { t ∈ ϕ ( x ) : f ( x,t ) = g ( x ) } . (5) Interpretation: Think about an economic agent whose environment is de scribed by a vector x ∈ S. The agent’s set of actions is T, but when the environment is x, she is restricted to choose her action only from the sub set ϕ ( x ) . Her utility of action t is f ( x,t ) , when the environment is x. Her objective is to choose an action in ϕ ( x ) to maximize her utility. 1 We shall assume that the set T is compact. Correspondence ϕ is said to be continuous if it is lower hemicontinuous and upper hemicontinuous. These are defined as follows: (LHC) ϕ is lower hemicontinuous at x if for every sequence { x n } in S converging to x and every t ∈ ϕ ( x ) , there exists a sequence { t n } in T such that t n ∈ ϕ ( x n ) and { t n } converges to t....
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 Fall '10
 riley

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