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hw1 - for any f R n R and D ³ R n Now suppose instead R R...

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ECON 401: Assignment 1 due date: September 9, 2011 1. Consider the following matrices A = 2 4 1 -2 1 2 4 0 4 2 1 3 5 ; B = 2 4 4 -2 3 1 -1 0 2 -1 0 3 5 ; and C = 2 6 6 4 1 1 -2 2 3 0 1 4 1 1 2 2 3 7 7 5 : (a) Calculate A + B , C ° A and C ° B: (b) Find the rank of A , B , C . (c) Find the determinant to of A , B . (d) Are A and B invertible? If no, why? If yes, °nd their inverse. 2. Find the supermum, in°mum, maximum and minimum of the set X in each of the following cases. (a) X = f x 2 [0 ; 1] : x is irrational g ; (b) X = ° x = 1 n : n 2 N ± ; where N denotes the set of natural numbers, i.e., N ± f 1 ; 2 ; 3 ; ::: g ; (c) X = ° x = 1 ² 1 n : n 2 N ± ; where N is de°ned above. (d) X = f x = z 2 : z 2 (0 ; 1] g . 3. Recall the following theorem. Theorem: suppose : R ! R is a strictly increasing function, i.e., x > y ) ( x ) > ( y ) . Then, arg max x 2D f ( x ) = arg max x 2D
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Unformatted text preview: )] for any f : R n ! R and D ³ R n . Now, suppose instead : R ! R is a weakly increasing function, i.e., x > y ) ( x ) ´ ( y ) . Do we still have arg max x 2D f ( x ) = arg max x 2D [ f ( x )]? If yes, prove it; if no, give a counterexample. 4. Give an example of an optimization problem with an in&nite number of solutions (i.e., an in&nite number of maximizer). 5. Find a function f : R ! R and a collection of sets S k ³ R such that f attains a maximum on each of S k , but not on [ 1 k =1 S k . 1...
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