Definitions and Some Theorems

2levelsuperiorinferiorsets

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This document was uploaded on 02/07/2014.

Ask a homework question - tutors are online