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Macroeconomics Exam Review 50

Macroeconomics Exam Review 50 - c 2001 Michael Carter All...

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there exists some x 𝑋 such that 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 𝑚 Therefore x 𝑋 ( p , 𝑚 ) which is nonempty. Let ˇ 𝑝 = min 𝑖 𝑝 𝑖 be the lowest price of the 𝑛 goods. Then 𝑋 ( p , 𝑚 ) 𝐵 ( 0 , 𝑚/ ˇ 𝑝 ) and so is bounded. (That is, no component of an affordable bundle can contain more than 𝑚/ ˇ 𝑝 units.) To show that 𝑋 ( p , 𝑚 ) is closed, let ( x 𝑛 ) be a sequence of consumption bundles in 𝑋 ( p , 𝑚 ). Since 𝑋 ( p , 𝑚 ) is bounded, x 𝑛 x 𝑋 . Furthermore 𝑝 1 𝑥 𝑛 1 + 𝑝 2 𝑥 𝑛 2 + ⋅ ⋅ ⋅ + 𝑝 𝑛 𝑥 𝑛 𝑛 𝑚 for every 𝑛 This implies that 𝑝 1 𝑥 1 + 𝑝 2 𝑥 2 + ⋅ ⋅ ⋅ + 𝑝 𝑛 𝑥 𝑛 𝑚 so that x 𝑛 x 𝑋 ( p , 𝑚 ). Therefore 𝑋 ( p , 𝑚 ) is closed. We have shown that 𝑋 ( p , 𝑚 ) is a closed and bounded subset of 𝑛 . Hence it is compact (Proposition 1.4). 1.232 Let x , y 𝑋 ( p , 𝑚 ). That is 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 𝑚 𝑛 𝑖 =1 𝑝 𝑖 𝑦 𝑖 𝑚 For any
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