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ps5 - false(20 points(a “Let x and y be n-dimensional...

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Atsushi Inoue ECG 765: Mathematics for Economists Fall 2009 Problem Set ] 5 Due in class on Tuesday, October 6 in class 1. Consider a lexicographic utility function U : < × < → < that satisfies two properties: (i) For any ( x, y ) and ( x 0 , y 0 ) such that x > x 0 , U ( x, y ) > U ( x 0 , y 0 ). (ii) For any ( x, y ) and ( x 0 , y 0 ) such that x = x 0 and y > y 0 , U ( x, y ) > U ( x 0 , y 0 ). (a) Prove that this utility function is not continuous. (2 points) (b) Suppose that the consumer maximizes this utility function U sub- ject to px + qy = I, x 0 , y 0 where p > 0, q > 0 and I > 0. Does a solution exist? If so, find the solution. (2 points) 2. True or false. Prove the statement if you claim that the statement is true. Give a specific counter example if you claim the statement is
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Unformatted text preview: false. (20 points) (a) “Let x and y be n-dimensional vectors. Then x · y = y · x .” (1 point) (b) “ AB = BA for any square matrices A and B .” (1 point) (c) Let X 2 denote XX . Then “( AB ) 2 = A 2 B 2 .” (1 point) (d) “If AB = B then A = I .” (1 point) 3. (a) Suppose that A and B are square matrices. Prove that ( AB ) k = A k B k if AB = BA . (1 point) (b) Show that ( AB ) k 6 = A k B k in general and conclude that ( A + B ) 2 does not equal A 2 + 2 AB + B 2 unless AB = BA . (1 point) 1...
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