pr2ec70204 - found through the solution to the social...

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Problem Set 2 Econ 702, Spring 2004 February 5, 2004 Problem 1 De f ning 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. Problem 2 In class, we de f ned the consumption possibility set in the following way, X = { x L : { k t +1 } t =0 0 such that (2) x 1 t +(1 δ ) k t k t +1 0 t (3) x 2 t [ 1 , 0] t x 3 t k t t k 0 = given } De f ne the consumption possiblity set with consumption. Problem 3 Show that the consumption possibility set, X ,andtheproduc t ion possibility set, Y are convex. Problem 4 Given the preferences, U ( x )= X t =0 β t u [ x 1 t +(1 δ ) k t k t +1 ]( 4 ) Verify the following, 1. Given u is continous, U is continous also. 2. Given u is strictly concave, U is strictly concave. 3. Given u is locally nonsatiated, U is locally nonsatiated. Problem 5 Show that the set of feasible allocations is compact ( X Y )
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Problem 6 We are trying to construct prices out of the allocations that we
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Unformatted text preview: found through the solution to the social planner’s problem. The allocations in the below problems are the optimal allocations from the social planner’s problem. We are trying to f nd the prices, p ∗ such that these allocations along with the prices constitute a valuation equilibrium. 1. Consumer’s problem, max x ∈ X U ( x ) ( 5 ) s.t. X t ( p ∗ 1 t x 1 t + p ∗ 2 t x 2 t + p ∗ 3 t x 3 t ) 2. Firm’s problem, max y ∈ Y X t ( p ∗ 1 t y 1 t + p ∗ 2 t y 2 t + p ∗ 3 t y 3 t ) ( 6 ) s.t. y 1 t = F ( − y 2 t , − y 3 t ) ∀ t Given this, derive a formula that links p ∗ 2 t , p ∗ 1 t and p ∗ 1 ,t +1 . Problem 7 Show that for the above problems where X, Y ⊂ L ⊂ R ∞ , First Order Conditions are the necessary conditions for optimality. 2...
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pr2ec70204 - found through the solution to the social...

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