solpr2ec70204

solpr2ec70204 - Problem Set 2 Econ 702, Spring 2004...

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Unformatted text preview: Problem Set 2 Econ 702, Spring 2004 Prepared by Ahu Gemici February 10, 2004 Problem 1 Defining the commodity space as a space of bounded real sequences, L = { { x it } ∞ t =0 , sup i,t x it < ∞ ∀ x } (1) Prove that L with the supnorm topology is a topological vector space. Solution: First we will show that L is a vector space and then endow it with a supnorm topology and show that L with the supnorm topology is a topological vector space. 1. To show that L is a vector space, it suffices to show that L is closed under vector addition and scalar multiplication. Take two sequences a = { a i } ∈ L and b = { b i } ∈ L . We first need to show that a + b ∈ L . ¯ a ≡ sup i a i < ∞ ¯ b ≡ sup i b i < ∞ ( because a, b ∈ L ) Let c = { c i } where c i = a i + b i ∀ i c i = a i + b i < ¯ a + ¯ b ≡ ¯ c < ∞ ⇒ c ∈ L (2) Now take k ∈ R + , k > 0. We need to show, a ∈ L ⇒ d = ka ∈ L ∀ k > (3) 1 Let ka i = d i ∀ i a i < ¯ a ∀ i ⇒ ka i < k ¯ a ∀ i ⇒ d i < k ¯ a ≡ ¯ d < ∞ ∀ i (4) By (2) and (4), L is a vector space. 2. Showing that L endowed with the supnorm topology is a topological vec- tor space. A topological vector space is a vector space which is endowed with a topol- ogy such that the maps ( x, y ) → x + y and ( λ, x ) → λx are continuous. So we have to show the continuity of the vector operations addition and scalar multiplication. Take ( x, y ) ∈ L s.t. x i → x and y i → y , then ( x i + y i )- ( x + y ) ≤ ( x i- x ) + ( y i- y ) → so x i + y i → x + y . Also, ( λx i- λx ) ≤ λ ( x i- x ) → so λx i → λx . Problem 2 In class, we defined the consumption possibility set in the following way, X = { x ∈ L : ∃{ k t +1 } ∞ t =0 ≥ such that (5) x 1 t + (1- δ ) k t- k t +1 ≥ ∀ t (6) x 2 t ∈ [ k t , 0] ∀ t x 3 t ∈ [- 1 , 0] ∀ t k = given } Define the consumption possiblity set with consumption....
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This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.

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solpr2ec70204 - Problem Set 2 Econ 702, Spring 2004...

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