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Macroeconomics Exam Review 243

Macroeconomics Exam Review 243 - c 2001 Michael Carter All...

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5.42 Assume that g satisfies the Quasiconvex CQ condition at x . That is, for every 𝑗 𝐵 ( x ), 𝑔 𝑗 is quasiconvex, 𝑔 𝑗 ( x ) = 0 and there exists ˆ x such that 𝑔 𝑗 x ) < 0. Consider the perturbation dx = ˆ x x . Quasiconvexity and regularity implies that for every binding constraint 𝑗 𝐵 ( x ) (Exercises 4.74 and 4.75) 𝑔 𝑗 x ) < 𝑔 𝑗 ( x ) = ⇒ ∇ 𝑔 𝑗 ( x ) 𝑇 x x ) = 𝑔 𝑗 ( x ) 𝑇 dx < 0 That is 𝐷𝑔 𝑗 [ x ]( dx ) < 0 Therefore, dx 𝐿 0 ( x ) = and g satisfies the Cottle constraint qualification condition. 5.43 If the binding constraints 𝐵 ( x ) are regular at x , their gradients are linearly independent. That is, there exists no 𝜆 𝑗 = 0 , 𝑗 𝐵 ( x ) such that 𝑗 𝐵 ( x ) 𝜆 𝑗 𝑔 𝑗 [ x ] = 0 By Gordan’s theorem (Exercise 3.239), there exists dx ∈ ℜ 𝑛 such that 𝑔 𝑗 [ x ] 𝑇 dx < 0 for every 𝑗 𝐵 ( x ) Therefore dx 𝐿 0 ( x ) = . 5.44 If 𝑔 𝑗 concave, 𝐵 𝑁 ( x ) = , and AHUCQ is trivially satisfied (with dx = 0 𝐿 1 ). For every 𝑗 , let 𝑆 𝑗 = { dx :
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