Macroeconomics Exam Review 90

# Macroeconomics Exam Review 90 - Solutions for Foundations...

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2. The barycentric coordinates of vertex x ? are ? ? =1w ith ? ± =0forevery ± = ² . Therefore the rule assigns vertex x ? the label ² . 3. Similarly, if x belongs to a proper face of ³ , it coordinates relative to the vertices not in that face are 0, and it cannot be assigned a label corresponding to a vertex not in the face. To be concrete, suppose that x conv { x 1 , x 2 , x 4 } . Then x = ? 1 x 1 + ? 2 x 2 + ? 4 x 4 ,? 1 + ? 2 + ? 4 =1 and ? ? =0for ²/ ∈{ 1 , 2 , 4 } . Therefore x 7−→ min { ² : ´ ? ? ? =0 }∈{ 1 , 2 , 4 } 2.129 1. Since ³ is compact, it is bounded (Proposition 1.1) and therefore it is contained in a suﬃciently large simplex µ . 2. By Exercise 3.74, there exists a continuous retraction : µ ³ . The composition · : µ ³ µ . Furthermore as the composition of continuous functions, · is continuous (Exercise 2.72). Therefore · has a ±xed point x µ ,thatis · ( x )= x . 3. Since · ( x ) ³ for every x µ ,wemusthave · ( x x ³ . Therefore, ( x x which implies that · ( x
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