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Macroeconomics Exam Review 236

Macroeconomics Exam Review 236 - c 2001 Michael Carter All...

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so that 𝑥 1 = 𝛼 𝛼 + (1 𝛼 ) 𝑚 𝑝 1 From the budget constraint (5.92) 𝑥 2 = (1 𝛼 ) 𝛼 + (1 𝛼 ) 𝑚 𝑝 2 5.26 The Lagrangean is 𝐿 ( x , 𝜆 ) = 𝑥 𝛼 1 1 𝑥 𝛼 2 2 . . . 𝑥 𝛼 𝑛 𝑛 𝜆 ( 𝑝 1 𝑥 1 + 𝑝 2 𝑥 2 + ... + 𝑝 𝑛 𝑥 𝑛 ) The first-order conditions for a maximum are 𝐷 𝑥 𝑖 𝐿 = 𝛼 𝑖 𝑥 𝛼 1 1 𝑥 𝛼 2 2 . . . 𝑥 𝛼 𝑖 1 𝑖 . . . 𝑥 𝛼 𝑛 𝑛 𝜆𝑝 𝑖 = 𝛼 𝑖 𝑢 ( 𝑥 ) 𝑥 𝑖 𝜆𝑝 𝑖 = 0 or 𝛼 𝑖 𝑢 ( 𝑥 ) 𝜆 = 𝑝 𝑖 𝑥 𝑖 𝑖 = 1 , 2 , . . . , 𝑛 (5.93) Summing over all goods and using the budget constraint 𝑛 𝑖 =1 𝛼 𝑖 𝑢 ( 𝑥 ) 𝜆 = 𝑢 ( 𝑥 ) 𝜆 𝑛 𝑖 =1 𝛼 𝑖 = 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 = 𝑚 Letting 𝑛 𝑖 =1 𝛼 𝑖 = 𝛼 , this implies 𝑢 ( x ) 𝜆
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