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Macroeconomics Exam Review 51

# Macroeconomics Exam Review 51 - Solutions for Foundations...

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3. Assume is locally nonsatiated. Then, for every x 𝑋 , there exists some y 𝑋 such that y x . Therefore, there is no best element. 1.234 Assume otherwise, that is assume that x x for every x 𝐵 ( p , 𝑚 ) but that 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 < 𝑚 . Let 𝑟 = 𝑚 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 be the unspent income. Spending the residual on good 1, the commodity bundle x = x + 𝑟 𝑝 1 e 1 is affordable 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 = 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 + 𝑝 1 𝑟 𝑝 1 = 𝑚 Moreover, since x x , x x , which contradicts the assumption that x is the best element in 𝑋 ( p , 𝑚 ). 1.235 Assume otherwise, that is assume that x x for every x 𝐵 ( p , 𝑚 ) but that 𝑛 𝑖 =1 𝑝 𝑖 𝑥 𝑖 < 𝑚 . This implies that x int 𝑋 ( p , 𝑚 ). Therefore, there exists a neigh- borhood 𝑁 of x with 𝑁 𝑋 ( p , 𝑚 ). Within this neighborhood, there exists some x 𝑁 𝑋 ( p , 𝑚 ) with x x
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