Macroeconomics Exam Review 240

# Macroeconomics Exam Review 240 - c 2001 Michael Carter All...

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with ? + ? ± ? ( x )= 0 and ? ? ± ? ( x 0 ² =1 , 2 ,...,³ DeFning ? ? = ? + ? ? ? , (5.100) can be written as ´µ ( x ? ? ´± ? [ x ] which is the Frst-order condition for an equality constrained problem. ±urthermore, if x satisFes the inequality constraints ± ( x ) 0 and ± ( x ) 0 it satisFes the equality ± ( x 0 5.34 Suppose that x solves the problem max x c ± x subject to x 0 with Lagrangean · = c ± x ? ± x Then there exists ? 0 such that ´ x · = c ± ? ± = 0 that is, ± ? = c . Conversely, if there is no solution, there exists x such that x 0 and c ± x > c ± 0 =0 5.35 There are two binding constraints at (4 , 0), namely ± ( ¸ 1 2 ¸ 1 + ¸ 2 4 ( ¸ 1 2 ¸ 2 0 with gradients ± (4 , 0) = (1 , 1) (4 , 0) = (0 , 1) which are linearly independent. Therefore the binding constraints are regular at (0
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