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Macroeconomics Exam Review 252

# Macroeconomics Exam Review 252 - c 2001 Michael Carter All...

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6.8 Supermodularity of Π( x , 𝑝, w ) follows from Exercises 2.50 and 2.51. To show strictly increasing differences, consider two price vectors w 2 w 1 Π( x , 𝑝, w 1 ) Π( x , 𝑝, w 2 ) = 𝑛 𝑖 =1 ( 𝑤 1 𝑖 ) 𝑥 𝑖 𝑛 𝑖 =1 ( 𝑤 2 𝑖 ) 𝑥 𝑖 = 𝑛 𝑖 =1 ( 𝑤 2 𝑖 𝑤 1 𝑖 ) 𝑥 𝑖 Since w 2 w 1 , w 2 w 1 0 and 𝑛 𝑖 =1 ( 𝑤 2 𝑖 𝑤 1 𝑖 ) 𝑥 𝑖 is strictly increasing in x . 6.9 For any 𝑝 2 𝑝 1 , 𝑦 2 = 𝑓 ( 𝑝 2 ) 𝑓 ( 𝑝 1 ) = 𝑦 1 and 𝑐 ( 𝑦 1 , 𝜃 ) 𝑐 ( 𝑦 2 , 𝜃 ) is increasing in 𝜃 and therefore ( 𝑐 ( 𝑓 ( 𝑝 2 ) , 𝜃 ) 𝑐 ( 𝑓 ( 𝑝 1 ) , 𝜃 )) is increasing in 𝜃 . 6.10 The firm’s optimization problem is max 𝑦 ∈ℜ + 𝜃𝑝𝑦 𝑐 ( 𝑦 ) The objective function 𝑓 ( 𝑦, 𝑝, 𝜃 ) = 𝜃𝑝𝑦 𝑐 ( 𝑦 ) is supermodular in 𝑦 (Exercise 2.49) displays strictly increasing differences in ( 𝑦, 𝜃 ) since 𝑓 ( 𝑦 2 , 𝑝, 𝜃 ) 𝑓 ( 𝑦 1 , 𝑝, 𝜃 ) = 𝜃𝑝 ( 𝑦 2 𝑦 1 ) ( 𝑐 ( 𝑦 2 ) 𝑐 ( 𝑦 1 ) ) is strictly increasing in 𝜃 for 𝑦 2 > 𝑦 1 . Therefore (Corollary 2.1.2), the firm’s output correspondence is strongly increasing and
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