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Unformatted text preview: 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 twosector 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 k x k < } 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 ) 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 = 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: agemici@ssc.upenn.edu and okp@ssc.upenn.edu. 1 Consumers problem: max X t X 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. X t X h t 3 X i =1 [ p A it ( h t ) x A it ( h t ) + p B it ( h t ) x B it ( h t )] Producers problem: max y A Y A ,y B Y B 3 X i =1 [ p A it ( h t ) y A it ( h t ) + p B it ( h t ) y B it ( h t )] An ArrowDebreu Competitive Equilibrium is ( p A * ,x A * ,y A * ,p B * ,x B * ,y B * ) such that 1. x A * ,x B * solves the consumers problem. 2. y A * ,y B * solves the firms 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 ) x + x for any (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 QuasiEquilibrium ((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 problem in recursive formulation;...
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