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Macroeconomics Exam Review 36

Macroeconomics Exam Review 36 - c 2001 Michael Carter All...

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𝑥 1 𝑥 2 𝑥 1 𝑥 2 𝑥 1 𝑥 2 3. A convex cone 1. A non-convex cone 2. A convex set Figure 1.6: A cone which is not convex, a convex set and a convex cone 1.184 See Figure 1.6. 1.185 Let x = ( 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ) belong to 𝑛 + , which means that 𝑥 𝑖 0 for every 𝑖 . For every 𝛼 > 0 𝛼 x = ( 𝛼 x 1 , 𝛼 x 2 , . . . , 𝛼 x 𝑛 ) and 𝛼𝑥 𝑖 0 for every 𝑖 . Therefore 𝛼 x ∈ ℜ 𝑛 + . 𝑛 + is a cone in 𝑛 . 1.186 Assume 𝛼 x + 𝛽 y 𝑆 for every x , y 𝑆 and 𝛼, 𝛽 ∈ ℜ + (1.26) Letting 𝛽 = 0, this implies that 𝛼 x 𝑆 for every x 𝑆 and 𝛼 ∈ ℜ + so that 𝑆
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