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Macroeconomics Exam Review 38

Macroeconomics Exam Review 38 - Solutions for Foundations...

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1.191 1. Suppose that y 𝑌 . Then, there exist 𝛼 , 𝛼 2 , . . . , 𝛼 8 0 such that y = 8 𝑖 =1 𝛼 𝑖 y 𝑖 and for the first commodity y 1 = 8 𝑖 =1 𝛼 𝑖 𝑦 𝑖 1 If y = 0, at least one of the 𝛼 𝑖 > 0 and hence y 1 < 0 since 𝑦 𝑖 1 < 0 for 𝑖 = 1 , 2 , . . ., 8. 2. Free disposal requires that y 𝑌, y y = y 𝑌 . Consider the production plan y = ( 2 , 2 , 2 , 2). Note that y y 3 and y y 6 . Suppose that y 𝑌 . Then there exist 𝛼 1 , 𝛼 2 , . . . , 𝛼 8 0 such that y = 8 𝑖 =1 𝛼 𝑖 y 𝑖 For the third commodity (component), we have 4 𝛼 1 + 3 𝛼 2 + 3 𝛼 3 + 3 𝛼 4 + 12 𝛼 5 2 𝛼 6 + 5 𝛼 8 = 2 (1.27) and for the fourth commodity 2 𝛼 2 1 𝛼 3 + 1 𝛼 4 + 5 𝛼 6 + 10 𝛼 7 2 𝛼 8 = 2 (1.28) Adding to (1.28) to (1.27) gives 4 𝛼 1 + 5 𝛼 2 + 2 𝛼 3 + 4 𝛼 4 + 12 𝛼 5 + 3 𝛼 6 + 10 𝛼 7 + 3 𝛼 8 = 4 which is impossible given that 𝛼 𝑖 0. Therefore, we conclude that
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