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Macroeconomics Exam Review 185

# Macroeconomics Exam Review 185 - c 2001 Michael Carter All...

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such that 𝐶 x 0 has no solution. Assume that 𝐵 is 𝑘 × 𝑛 and 𝐶 is 𝑙 × 𝑛 where 𝑙 = 𝑚 𝑘 . Applying Stiemke’s theorem to 𝐶 , there exists z > 0, z ∈ ℜ 𝑙 . Define y ∈ ℜ 𝑚 + by 𝑦 𝑗 = { 0 𝑗 = 1 , 2 , . . . , 𝑘 𝑦 𝑗 = 𝑧 𝑗 𝑘 𝑗 = 𝑘 + 1 , 𝑘 + 2 , . . . , 𝑚 Then x , y is the desired solution since for every 𝑗 , 𝑗 = 1 , 2 , . . . , 𝑚 either 𝑦 𝑗 > 0 or ( 𝐴 x ) 𝑗 = ( 𝐵 x ) 𝑗 > 0. 3.248 Consider the dual pair ( 𝐴 𝐼 ) x 0 and ( 𝐴 𝑇 , 𝐼 ) ( y z ) = 0 , y 0 , z 0 By Tucker’s theorem, this has a solution x , y , z such that 𝐴 x 0 , x 0 , 𝐴 𝑇 y + z = 0 , y 0 , z 0 𝐴 x + y > 0 𝐼 x + 𝐼 z > 0 Substituting z = 𝐴 𝑇 y implies 𝐴 𝑇 y 0 and x 𝐴 𝑇 y > 0 3.249 Consider the dual pair 𝐴 x 0 and 𝐴 𝑇 y = 0 , y 0 where 𝐴 is an 𝑚 × 𝑛 matrix. By Tucker’s theorem, there exists a pair of solutions x ∈ ℜ 𝑛 and y ∈ ℜ 𝑚 such that 𝐴 x + y > 0 (3.77) Assume that 𝐴 x > 0 has no solution (Gordan I). Then there exists some
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