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Macroeconomics Exam Review 251

# Macroeconomics Exam Review 251 - c 2001 Michael Carter All...

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Using Young’s theorem (Theorem 4.2), 𝐷 𝑦 𝑥 𝑖 [ w , 𝑦 ] = 𝐷 2 𝑤 𝑖 𝑦 𝑐 [ w , 𝑦 ] = 𝐷 2 𝑦𝑤 𝑖 𝑐 [ w , 𝑦 ] = 𝐷 𝑤 𝑖 𝐷 𝑦 𝑐 [ w , 𝑦 ] Therefore 𝐷 𝑦 𝑥 𝑖 [ w , 𝑦 ] 0 ⇐⇒ 𝐷 𝑤 𝑖 𝐷 𝑦 𝑐 [ w , 𝑦 ] 0 6.7 The demand functions must satisfy the budget contraint identically, that is 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 ( p , 𝑚 ) = 𝑚 for every p and 𝑚 Differentiating with respect to m 𝑛 𝑖 =1 𝑝 𝑖 𝐷 𝑚 𝑥 𝑖 [ p , 𝑚 ] = 1 This is the Engel aggregation condition, which simply states that any additional income be spent on some goods. Multiplying each term by 𝑥 𝑖 𝑚/ ( 𝑥 𝑖 𝑚 ) 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 𝑚 𝑚 𝑥 𝑖 ( p , 𝑚 ) 𝐷 𝑚 𝑥 𝑖 [ p , 𝑚 ] = 1 the Engel aggregation condition can be written in elasticity form 𝑛 𝑖 =1 𝛼 𝑖 𝜂 𝑖 = 1 where 𝛼 𝑖 = 𝑝 𝑖 𝑥 𝑖 /𝑚 is the budget share of good 𝑖 . On average, goods must have unit income elasticities.
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