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 â‰¿ x 1 and let = { x 1 , x 2 } . Assume that ( x 2 , 1 ) â‰¥ ( x 1 , 1 ). Then x 2 âˆˆ ( 1 , ) which implies that x 2 âˆˆ ( 2 , ) for any 2 â‰¿ 1 . (If x 1 âˆˆ ( 2 , ), then also x 1 âˆ¨ x 2 = x 2 âˆˆ ( 2 , ) since is increasing in ( , ).) But this implies that ( x 2 , 2 ) â‰¥ ( x 1 , 2 ). satisfies the single crossing condition. 2.66 First, assume that is continuous. Let be an open subset in and = âˆ’ 1 ( ). If = âˆ… , it is open. Otherwise, choose âˆˆ and let = ( ) âˆˆ . Since is open, there exists a neighborhood ( ) âŠ† . Since is continuous, there exists a corresponding neighborhood ( ) with ( ( )) âŠ† ( ( )). Since ( ( )) âŠ† , ( ) âŠ† . This establishes that for every âˆˆ there exist a neighborhood ( ) contained in . That is,....
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