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Macroeconomics Exam Review 137

Macroeconomics Exam Review 137 - Solutions for Foundations...

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from 𝐴 x = c to give 𝐴 x = 𝐴 ( x 𝑡 y ) = c (c) Note that x > 0 and therefore ˆ 𝑡 > 0 which implies that ˆ 𝑥 𝑗 > 0 for every 𝑦 𝑗 0. For every 𝑦 𝑗 > 0, 𝑥 𝑗 /𝑦 𝑗 ˆ 𝑡 , which implies that 𝑥 𝑗 ˆ 𝑡𝑦 𝑗 , so that ˆ 𝑥 𝑗 𝑥 𝑗 ˆ 𝑡𝑦 𝑗 0 Therefore, ˆ x is a feasible solution. (d) There exists some coordinate such that ˆ 𝑡 = 𝑥 /𝑦 so that ˆ 𝑥 = 𝑥 ˆ 𝑡𝑦 = 0 so that c = 𝑘 𝑗 𝑗 = =1 ˆ 𝑥 𝑗 𝐴 𝐽 ˆ 𝑥 is a feasible solution with one less positive component. 3. Starting with any nonbasic feasible solution, this elimination technique can be repeated until the remaining vectors are linearly independent and a basic feasible solution is obtained. 3.112 1. Exercise 1.173. 2. For each 𝑖
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