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Macroeconomics Exam Review 247

# Macroeconomics Exam Review 247 - c 2001 Michael Carter All...

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which implies that 𝛼𝑧 𝝀 𝑇 0 𝛼 ( 𝑧 + 1) 𝝀 𝑇 0 or 𝛼 0. Similarly, let { e 1 , e 2 , . . . , e 𝑚 } denote the standard basis for 𝑚 (Example 1.79). For any 𝑗 = 1 , 2 , . . . , 𝑚 , the point ( 0 e 𝑗 , 𝑧 ) (which corresponds to decreasing resource 𝑗 by one unit) belongs to 𝐵 and therefore (from (5.75)) 𝑧 𝝀 𝑇 ( 0 e 𝑗 ) = 𝑧 + 𝜆 𝑗 𝑧 𝝀 𝑇 0 = 𝑧 which implies that 𝜆 𝑗 0. 5.51 Let ˆ c = 𝑔 x ) < 0 and ˆ 𝑧 = 𝑓 x ) Suppose 𝛼 = 0. Then, since 𝐿 is nonzero, at least one component of 𝝀 must be nonzero. That is, 𝝀 0 and therefore 𝝀 𝑇 ˆ 𝑐 < 0 (5.107) But (ˆ c , ˆ 𝑧 ) 𝐴 and (5.74) implies 𝛼 ˆ 𝑧 𝝀 𝑇 ˆ c 𝛼𝑧 𝝀 𝑇 0 and therefore 𝛼 = 0 implies 𝝀 𝑇 ˆ 𝑐 0 contradicting (5.107). Therefore, we conclude that 𝛼 > 0. 5.52 The utility’s optimization problem is max 𝑦,𝑌 0 𝑆 ( 𝑦, 𝑌 ) = 𝑛 𝑖 =1 𝑦 𝑖 0 ( 𝑝 𝑖 ( 𝜏 ) 𝑐 𝑖 ) 𝑑𝜏 𝑐 0 𝑌 subject to 𝑔 𝑖 ( y , 𝑌 ) = 𝑦 𝑖 𝑌 0 𝑖 = 1 , 2 , . . ., 𝑛 The demand independence assumption ensures that the objective function 𝑆 is concave,
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