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prelsp04_sol_total

prelsp04_sol_total - Suggested Solutions for Econ 702...

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Suggested Solutions for Econ 702 Qualifying Exam, Spring 2004 Prepared by Ahu Gemici and Omer Kagan Parmaksiz * August 18, 2004 Growth Model Question 1 This is a two-sector growth model. There is an investment good sector and a consumption good sector. Commodity Space: L = { x | x t ( h t ) = ( x 1 t ( h t ) , x 2 t ( h t ) , x 3 t ( h t )) R 3 t, h t and x < ∞} Consumption possibility set: —– X = { x A , x B L : ∃{ c A t ( h t ) , c B t ( h t ) , k t +1 ( h t ) } t =0 0 such that c A t ( h t ) + k t +1 ( h t ) = x A 1 t ( h t ) + (1 - δ ) k t ( h t - 1 ) t, h t c B t ( h t ) = x B 1 t ( h t ) t, h t x A 2 t ( h t ) + x B 2 t ( h t ) [0 , 1] t, h t x A 3 t ( h t ) + x B 3 t ( h t ) k t ( h t - 1 ) t, h t k 0 = given } where x A 1 t ( h t )=received goods in terms of apples, x A 2 t ( h t )=labor supply to tech- nology A, x A 3 t ( h t )=capital service to technology A, x B 1 t ( h t )=received goods in terms of bananas, x B 2 t ( h t )=labor supply to technology B, x B 3 t ( h t )=capital ser- vice to technology B. Production possibility sets: Y A = { y L : y A 1 t ( h t ) F A ( y A 3 t ( h t ) , y A 2 t ( h t )) } Y B = { y L : y B 1 t ( h t ) F B ( y B 3 t ( h t ) , y B 2 t ( h t )) } * Email: [email protected] and [email protected] 1
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Consumer’s problem: max t h t u [ c A t ( h t ) , c B t ( h t ) , 1 - x A 2 t ( h t ) + 1 - x B 2 t ( h t )] s.t. t h t 3 i =1 [ p A it ( h t ) x A it ( h t ) + p B it ( h t ) x B it ( h t )] Producer’s problem: max y A Y A ,y B Y B 3 i =1 [ p A it ( h t ) y A it ( h t ) + p B it ( h t ) y B it ( h t )] An Arrow-Debreu Competitive Equilibrium is ( p A * , x A * , y A * , p B * , x B * , y B * ) such that 1. x A * , x B * solves the consumer’s problem. 2. y A * , y B * solves the firm’s problem. 3. Markets clear, x A * = y A * x B * = y B * Question 2 Theorem 1 (FBWT) If the preferences of consumers are nonsatiated ( ∃{ x n } ∈ X that converges to x X such that U ( x n ) > U ( x )), an allocation ( x * , y * ) of an ADE ( p * , x * , y * ) is PO. Theorem 2 (SBWT) If (i) X is convex, (ii) preference is convex (for x, x X, if x < x , then x < (1 - θ ) x + θx for any θ (0 , 1)), (iii) U ( x ) is continuous, (iv) Y is convex, (v)Y has an interior point, then with any PO allocation ( x * , y * ) such that x * is not a satuation point, there exists a continuous linear functional p * such that ( x * , y * , p * ) is a Quasi-Equilibrium ((a) for x X which U ( x ) U ( x * ) implies p * ( x ) p * ( x * ) and (b) y Y implies p * ( y ) p * ( y * )) Question 3 First write down the problem of the household for future reference. The ag- gragate state variables are technology shock vector z = ( z A , z B ) capital stock vector of two sectors K = ( K A , K B ) and individual state variable is asset hold- ings of the household. Note that the stochastic processes are independent. The 2
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problem in recursive formulation; V ( a, z, K ) = max c A ,c B ,a ,n ( φ ( c A , c B ) 1 - σ 1 - σ + α (1 - n )) + β z Γ A Γ B V ( a , z , K ) s.t. c A + p B c B + a = wn + Ra given, R = z A F 1 ( K A , N A ) + 1 - δ = ( z B F 1 ( K B , N B ) + 1 - δ ) p B w = z A F 2 ( K A , N A ) = z B F 2 ( K B , N B ) p B K A = G A ( K, z ) , K B = G B ( K, z ) N A = H A ( K, z ) , N B = H B ( K, z ) , p B = p B ( z, K ) Note that we are in a representative agent environment and complete market setup is equivalent to closing markets down. Also the numeriare good is apples.
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