2005ps_solutions

2005ps_solutions - Suggested Solutions to Problem Set 1...

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Suggested Solutions to Problem Set 1 Econ 702, Spring 2005 January 24, 2005 Problem 1 For a representative agent economy prove the following: x * PO ( ε ) x * arg max x X u ( x ) (1) Solution: 1. First showing Assume this is an economy with a large (finite) number of identical agents. Suppose x * ( ε ) but x * i 6∈ arg max x i X u ( x i ) for some i , where i is the index for identical agents. ⇒ ∃ ˜ x i s.t. u x i ) > u ( x * i ) and ˜ x i X Construct the following allocation, ˜ x = ( x * 1 , x * 2 , .... , ˜ x j , .... ) The allocation ˜ x is feasible and it gives more utility to agent j while keeping all the others with the same utility.This contradicts with the fact that x * is Pareto optimal. 2. Showing Suppose x * i arg max x i X u ( x i ) i but x * 6∈ ( ε ) Since x * 6∈ ( ε ), a feasible allocation ˜ x such that, u x i ) u ( x * i ) for all i u x i ) > u ( x * i ) for some i (2) But this means that for some i x * i 6∈ arg max x i X u ( x i ). Contradiction. Problem 2 Consider the following social planner’s problem: max { c t ,l t ,n t ,k t +1 } X t =0 β t u ( c t , ` t ) (3) 1
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s.t. c t + k t +1 f ( k t , n t ) k 0 given, c t , k t +1 , n t , ` t 0 ` t + n t = 1 Show that the set of feasible allocations is convex. Solution: 1. Showing that the constraint set is convex. Define, B = {{ c t , l t , k t +1 } t =0 ` : c t + k t +1 f ( k t , 1 - l t ) , c t 0 and k t +1 0 t } Let { c 1 t , l 1 t , k 1 t +1 } ∈ B , { c 2 t , l 2 t , k 2 t +1 } ∈ B and θ [0 , 1] . The following holds, c 1 t + k 1 t +1 = f ( k 1 t , 1 - l 1 t ) (4) c 2 t + k 2 t +1 = f ( k 2 t , 1 - l 2 t ) (5) Then for 0 < θ < 1, [ θc 1 t + (1 - θ ) c 2 t ] + [ θk 1 t +1 + (1 - θ ) k 2 t +1 ] = θf ( k 1 t , 1 - l 1 t ) + (1 - θ ) f ( k 2 t , 1 - l 2 t ) (6) Assuming f is concave as usual, θf ( k 1 t , 1 - l 1 t ) + (1 - θ ) f ( k 2 t , 1 - l 2 t ) f ( θk 1 t + (1 - θ ) k 2 t , θ (1 - l 1 t ) + (1 - θ )(1 - l 2 t )) (7) (6) and (7) imply, [ θc 1 t + (1 - θ ) c 2 t ] + [ θk 1 t +1 + (1 - θ ) k 2 t +1 ] f ( θk 1 t + (1 - θ ) k 2 t , θ (1 - l 1 t ) + (1 - θ )(1 - l 2 t )) (8) ⇒ { (1 - θ ) c 1 t + θc 2 t , (1 - θ ) l 1 t + θl 2 t , (1 - θ ) k 1 t +1 + θk 2 t +1 } ∈ B ⇒ B , the set of sequences { c t , l t , k t +1 } t =0 that satisfy the constraints is a convex subset of R . Problem 3 Consider the social planner’s problem (SPP)above with Cobb-Douglas technology, partial depreciation and CRRA preferences. Derive the Euler equa- tion. 2
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Solution: The particular functional forms are; f ( k t , n t ) = k θ t n (1 - θ ) t u ( c t , l t ) = ( c 1 - σ t 1 - σ ) + ( l 1 - γ t 1 - γ ) Letting; V ( k 0 ) = max { c t ,l t ,n t ,k t +1 }∈ C X t =0 β t u ( c t , ` t ) where C is the constraint set then after substituting for the consumption from the feasibility constraint,the generic Euler equation is; dV ( k 0 ) dk t +1 = u c ( c t , l t ) - βu c ( c t +1 , l t +1 )[1 - δ + f k ( k t +1 , n t +1 )] = 0 where the derivatives are; u c ( c t , l t ) = c - σ t f k ( k t , n t ) = θk θ - 1 t n 1 - θ t substituting Problem 4 Defining the commodity space as a space of bounded real sequences, L = { { ` it } t =0 , sup i,t | ` it | < ∞ ∀ ` } (9) Prove that L endowed with the supnorm is a topological vector space (TVS).
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2005ps_solutions - Suggested Solutions to Problem Set 1...

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