Definitions and Some Theorems


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Unformatted text preview: trictly concave if the above holds with strict inequality for all ∈ 0,1 . Likewise, the definition of a convex function (strictly convex function) is similarly defined: The function is said to be convex if, for all and in , any convex combination of and , ≡ 1 , satisfies 1 , ∀ ∈ 0,1 . 2. Level, Superior, Inferior Sets Level, Superior, and Inferior Sets Economics, as in other disciplines that use mathematics, pays particular close attention to real‐valued functions (functions that map into . An important concept, which you may recognize from micro, is the level ≡ ⋮∈, , where ∈ ⊂ . In words, this is the set: set of all values in the doman of that equals some number in the range . We say that is a level set relative to . We have similar concepts called superior sets and inferior sets : Axiom 4: Strict Monotonicity For all ∈ ≫ , then ≻ . Axiom 5: Strict Convexity If , , ∈ , if and ≿ , then , then ≿ , and if ≻ for all Utility Function: A real valued function u: → is called a utility function representing the preference relation ≿ if for all ∈ ,∈ , u( ⟺ ≿ Proposition A: If a preference relation ≿ is complete, transitive, continuous, and strictly monotonically increasing, there exists a continuous real valued function u: → which represents ≿ . Proposition B: If a preference relation ...
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This document was uploaded on 02/07/2014.

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