Macroeconomics Exam Review 74

Macroeconomics Exam Review 74 - To show that satisfies the...

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Unformatted text preview: To show that satisfies the single crossing condition, choose any x 2 ≿ x 1 and let = { x 1 , x 2 } . Assume that ( x 2 , 1 ) ≥ ( x 1 , 1 ). Then x 2 ∈ ( 1 , ) which implies that x 2 ∈ ( 2 , ) for any 2 ≿ 1 . (If x 1 ∈ ( 2 , ), then also x 1 ∨ x 2 = x 2 ∈ ( 2 , ) since is increasing in ( , ).) But this implies that ( x 2 , 2 ) ≥ ( x 1 , 2 ). satisfies the single crossing condition. 2.66 First, assume that is continuous. Let be an open subset in and = − 1 ( ). If = ∅ , it is open. Otherwise, choose ∈ and let = ( ) ∈ . Since is open, there exists a neighborhood ( ) ⊆ . Since is continuous, there exists a corresponding neighborhood ( ) with ( ( )) ⊆ ( ( )). Since ( ( )) ⊆ , ( ) ⊆ . This establishes that for every ∈ there exist a neighborhood ( ) contained in . That is,....
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