Macroeconomics Exam Review 149

Macroeconomics Exam Review 149 - Solutions for Foundations...

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so that 𝑓 is quasiconcave. The second inequality implies that 𝑓 ( 𝛼𝑥 1 + (1 𝛼 ) 𝑥 2 ) max { 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) } = 𝑓 ( 𝑥 2 ) so that 𝑓 is also quasiconvex. Conversely, if 𝑓 is decreasing 𝑓 ( 𝑥 1 ) 𝑓 ( 𝛼𝑥 1 + (1 𝛼 ) 𝑥 2 ) 𝑓 ( 𝑥 2 ) for every 0 𝛼 1. The first inequality implies that 𝑓 ( 𝑥 1 ) = max { 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) } ≥ 𝑓 ( 𝛼𝑥 1 + (1 𝛼 ) 𝑥 2 ) so that 𝑓 is quasiconvex. The second inequality implies that 𝑓 ( 𝛼𝑥 1 + (1 𝛼 ) 𝑥 2 ) max { 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) } = 𝑓 ( 𝑥 2 ) so that 𝑓 is also quasiconcave. 3.146 𝑓 ( 𝑐 ) = { x 𝑋 : 𝑓 ( x ) 𝑎 } = { x 𝑋 : 𝑓 ( x ) ≥ − 𝑐 } = 𝑓 ( 𝑐 ) 3.147 For given 𝑐 and 𝑚 , choose any p 1 and p 2 in 𝑣 ( 𝑐 ). For any 0 𝛼 1, let ¯ p = 𝛼 p 1 + (1 𝛼 ) p 2 . The key step is to show that any commodity bundle x which is affordable at ¯ p is also affordable at either p 1 or p 2 . Assume that x is affordable at ¯ p , that is x is in the budget set x 𝑋 p , 𝑚 ) = { x : ¯ px 𝑚 } To show that x is affordable at either p 1 or p 2 , that is x 𝑋
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