Problem_Set_6

# Problem_Set_6 - R deﬁned by f x = ax b x(1 x for all x...

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T.Mitra, Fall, 2008 Economics 6170 Problem Set 6 [For practice only; do not hand in solutions] 1. [Open Sets] (a) Let S and T be open sets in R n . Show that S T is also an open set in R n . (b) Let A be the set deﬁned by: A = { ( x 1 ,x 2 ) R 2 : x 1 > 0 ,x 2 > 0 ,x 1 x 2 > 1 } Express A as the intersection of two sets, and use (a) to show that A is open in R 2 . (c) Let B be the set deﬁned by: B = { ( x 1 ,x 2 ) R 2 : x 1 0 ,x 2 0 ,x 1 x 2 > 1 } Is B open in R 2 ? Explain. 2. [Closed Sets] (a) Let S and T be open sets in R n . Show that S T is also an open set in R n . (b) Let A be the set deﬁned by: A = { ( x 1 ,x 2 ) R 2 : x 1 0 ,x 2 0 ,x 1 + x 2 1 } and let B be the complement of A in R 2 . Express B as the union of three sets and show that B is open in R 2 . (c) Show that A is closed in R 2 . 3. [Continuity of Functions] Let f : R n R be a continuous function on R n , and let ¯ x be a vector in R n , satisfying f x ) > 0 . Show that there is a positive real number r such that: f ( x ) > 0 for all x B x,r ) 1

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4. [Extension of Weierstrass theorem] Let a and b be positive real numbers, and let f be a function from R + to
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Unformatted text preview: R + , deﬁned by: f ( x ) = ax + b [ x/ (1 + x )] for all x ≥ Consider the following constrained maximization problem: Maximize f ( x )-x subject to x ≥ ± ( P ) (a) If a < 1 , show that there exists a solution to problem ( P ) . (b) If a ≥ 1 , show that there is no solution to problem ( P ) . 5. [Extension of Weierstrass Theorem] Let p and q be arbitrary positive numbers, and let f : R 2 + → R be a continuous function on R 2 + . Suppose there is (¯ x 1 , ¯ x 2 ) ∈ R 2 + which satisﬁes f (¯ x 1 , ¯ x 2 ) = 1 . Consider the constrained minimization problem: Minimize px 1 + qx 2 subject to f ( x 1 ,x 2 ) ≥ 1 and ( x 1 ,x 2 ) ∈ R 2 + ( Q ) Show that there is a solution to problem ( Q ) . 2...
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## This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell.

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Problem_Set_6 - R deﬁned by f x = ax b x(1 x for all x...

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