Macroeconomics Exam Review 242

Macroeconomics Exam Review 242 - c 2001 Michael Carter All...

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5.39 Assume x 𝐷 ( x ). That is, there exists ¯ 𝛼 such that x + 𝛼 x 𝑆 for every 𝛼 [0 , ¯ 𝛼 ]. For 𝑘 = 1 , 2 , . . . , let 𝛼 𝑘 = ¯ 𝛼/𝑘 . Then x 𝑘 = x + 𝛼 𝑘 x 𝑆 , x 𝑘 x and ( x 𝑘 x ) /𝛼 𝑘 = ( x + 𝛼 𝑘 x x ) /𝛼 𝑘 = x . Therefore, x 𝑇 ( x ). 5.40 Let dx 𝑇 ( x ). Then there exists a feasible sequence { x 𝑘 } converging to x and a sequence { 𝛼 𝑘 } of nonnegative scalars such that the sequence { ( x 𝑘 x ) /𝛼 𝑘 } converges to dx . For any 𝑗 𝐵 ( x ), 𝑔 𝑗 ( x ) = 0 and 𝑔 𝑗 ( x 𝑘 ) = 𝐷𝑔 𝑗 [ x ]( x 𝑘 x ) + 𝜂 𝑗 x 𝑘 x where 𝜂 𝑗 0 as k → ∞ . This implies 1 𝛼 𝑘 𝑔 𝑗 ( x 𝑘 ) = 1 𝛼 𝑘 𝐷𝑔 𝑗 [ x ]( x 𝑘 x ) + 𝜂 𝑗 ( x 𝑘 x ) /𝛼 𝑘 Since x 𝑘 is feasible 1 𝛼 𝑘 𝑔 𝑗 ( x 𝑘 ) 0 and therefore 𝐷𝑔 𝑗 [ x ](( x 𝑘 x ) /𝛼 𝑘 ) + 𝜂 𝑖 ( x 𝑘 x ) /𝛼 𝑘 0 Letting 𝑘 → ∞
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