This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff

Click to edit the document details