Macroeconomics Exam Review 220

# Macroeconomics Exam Review 220 - c 2001 Michael Carter All...

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4.70 For 𝑆 ⊆ ℜ , 𝑓 ( 𝑥 ) = 𝑓 ( 𝑥 ) and (4.32) becomes ( 𝑓 ( 𝑥 2 ) 𝑓 ( 𝑥 1 )( 𝑥 2 𝑥 1 ) 0 for every 𝑥 1 , 𝑥 2 𝑆 . This is equivalent to 𝑓 ( 𝑥 2 )( 𝑥 2 𝑥 1 ) 𝑓 ( 𝑥 1 )( 𝑥 2 𝑥 1 )0 or 𝑥 2 > 𝑥 1 = 𝑓 ( 𝑥 2 ) 𝑓 ( 𝑥 1 ) 𝑓 is strictly convex if and only if the inequalities are strict. 4.71 𝑓 is increasing if and only if 𝑓 ′′ = 𝐷𝑓 0 (Exercise 4.35). 𝑓 is strictly increasing if 𝑓 ′′ = 𝐷𝑓 > 0 (Exercise 4.36). 4.72 Adapting the previous example 𝑓 ′′ ( 𝑥 ) = 𝑛 ( 𝑛 1) 𝑥 𝑛 2 = = 0 if 𝑛 = 1 0 if 𝑛 = 2 , 4 , 6 , 𝑑𝑜𝑡𝑠 indeterminate otherwise Therefore, the power function is convex if 𝑛 is even, and neither convex if 𝑛 3 is odd. It is both convex and concave when 𝑛 = 1. 4.73 Assume 𝑓 is quasiconcave, and 𝑓 ( x ) 𝑓 ( x 0 ). Differentiability at x 0 implies for all 0 < 𝑡 < 1 𝑓 ( x 0 + 𝑡 ( x x 0 ) = 𝑓 ( x 0 ) + 𝑓 ( x 0 ) 𝑡 ( x x 0 ) + 𝜂 ( 𝑡 ) 𝑡 ( x x 0 ) where 𝜂 ( 𝑡 ) 0 and 𝑡 0. Quasiconcavity implies 𝑓 ( x 0 + 𝑡 ( x x 0 ) 𝑓 ( x 0 ) and therefore 𝑓 ( x 0 ) 𝑡 ( x x 0 ) + 𝜂 ( 𝑡 ) 𝑡 ( x x 0 ) ∥ ≥ 0 Dividing by 𝑡 and letting 𝑡
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