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Macroeconomics Exam Review 172

# Macroeconomics Exam Review 172 - Solutions for Foundations...

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3. Let x 𝑆 . Then 𝑓 ( x ) 0 for every 𝑓 𝑆 so that x 𝑆 ∗∗ . 4. Exercise 1.79. 3.221 Let 𝑓 𝑆 2 . Then 𝑓 ( x ) 0 for every x 𝑆 2 . A fortiori , since 𝑆 1 𝑆 2 , 𝑓 ( x ) 0 for every x 𝑆 1 . Therefore 𝑓 𝑆 1 . 3.222 Exercise 3.220 showed that 𝑆 𝑆 ∗∗ . To show the converse, let y / 𝑆 . By Proposition 3.14, there exists some 𝑓 𝑋 and 𝑐 such that 𝑓 ( y ) > 𝑐 𝑓 ( x ) < 𝑐 for every x 𝑆 Since 𝑆 is a cone, 0 𝑆 and 𝑓 ( 0 ) = 0 < 𝑐 . Since 𝛼𝑆 = 𝑆 for every 𝛼 > 0 then 𝑓 ( x ) < 0 for every x 𝑆 so that 𝑓 𝑆 . 𝑓 ( y ) > 0, y / 𝑆 ∗∗ . That is y / 𝑆 = y / 𝑆 ∗∗ from which we conclude that 𝑆 ∗∗ 𝑆 . 3.223 Let 𝐾 = cone { 𝑔 1 , 𝑔 2 , . . . , 𝑔 𝑚 } = { 𝑔 𝑋 : 𝑔 = 𝑚 𝑗 =1 𝜆 𝑗 𝑔 𝑗 , 𝜆 𝑗 0 } be the set of all nonnegative linear combinations of the linear functionals 𝑔 𝑗 . 𝐾 is a closed convex cone. Suppose that 𝑓 / cone { 𝑔 1 , 𝑔 2 , . . . , 𝑔 𝑚 } , that is assume that 𝑓 / 𝐾 . Then {
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