homework8_sol

Homework8_sol - ECO 475 HW8 Solution Joon Song Assume the followings& u is strictly increasing and strictly concave& u is continuously

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Unformatted text preview: ECO 475 HW8 Solution Joon Song Nov 11, 2009 Assume the followings. & u ( ¡ ) is strictly increasing and strictly concave. & u is continuously di/erentiable and lim c & u ( c ) = 1 & F ( ¡ ; ¡ ) is strictly increasing, strictly concave and F ( k; 0) = F (0 ;n ) = 0 for any k ¢ and n ¢ . & F is continuously di/erentiable and lim k & F ( k;n ) = 1 for any n > and lim n & F n ( k;n ) = 1 for any k > . & F is homogeneous of degree 1. 1. Firms own capital and consumers own the &rms (1) ADE An ADE is a set of price f p t ;w t g 1 t =0 and allocation f c t ;k t +1 ;n t ;& t g 1 t =0 sequences such that 1. Consumer&s Problem Given f p t ;w t g 1 t =0 and f & t g 1 t =0 , f c t g 1 t =0 solves the following: max f c t g 1 X t =0 ¡ t u ( c t ) subject to 1 X t =0 p t c t = 1 X t =0 p t ( & t + w t ) : 2. Firm&s Problem Given f p t ;w t g 1 t =0 , f k t +1 ;n t ;& t g 1 t =0 solves the following: max f k t +1 ;n t ;& t g 1 X t =0 p t & t subject to & t = F ( k t ;n t ) + (1 £ ¢ ) k t £ k t +1 £ w t n t ; 8 t . k > is given. 3. Markets clear: 1 & (Labor Market) n t = 1 ; 8 t . & (Goods Market) c t + k t +1 = F ( k t ;n t ) + (1 ¡ & ) k t ; 8 t . Characterization of ADE From consumer&s problem, we have u ( c t ) = ¡u ( c t +1 ) p t p t +1 , 8 t . (1) From ¡rm&s problem, we get p t p t +1 = F k ( k t +1 ;n t +1 ) + 1 ¡ & , 8 t . (2) w t = F n ( k t ;n t ) , 8 t . (3) where n t = n t +1 = 1 using the equilibrium condition. Combining (1) and (2), we get u ( c t ) = ¡u ( c t +1 ) [ F k ( k t +1 ; 1) + 1 ¡ & ] , 8 t . (4) which is same as the EE from the problem where consumer holds capital. The following conditions jointly characterize the equilibrium. u ( c t ) = ¡u ( c t +1 ) [ F k ( k t +1 ; 1) + 1 ¡ & ] , 8 t . (5) p t p t +1 = F k ( k t +1 ; 1) + 1 ¡ & , 8 t . (6) w t = F n ( k t ; 1) , 8 t . (7) ¢ t = F ( k t ; 1) + (1 ¡ & ) k t ¡ k t +1 ¡ w t , 8 t . (8) n t = 1 , 8 t . (9) c t + k t +1 = F ( k t ; 1) + (1 ¡ & ) k t , 8 t . (10) (Consumer&s budget constraint is redundant by Walras&Law.) (2) SME A SME is a set of price f p t ;q t ;w t g 1 t =0 and allocation f c t ;k t +1 ;n t ;¢ t ;s t g 1 t =0 sequences such that 1. Consumer&s Problem Given f q t ;w t g 1 t =0 and f ¢ t g 1 t =0 , f c t ;s t +1 g 1 t =0 solves the following: max f c t ;s t +1 g 1 X t =0 ¡ t u ( c t ) 2 subject to c t + q t s t +1 = w t + s t & t + q t s t ; 8 t . s t +1 & & s; 8 t . s & & s is given. where & s < is a natural borrowing limit which in turn guarantees no Ponzi game scheme. 2. Firm&s Problem Given f p t ;w t g 1 t =0 , f k t +1 ;n t ;& t g 1 t =0 solves the following: max f k t +1 ;n t ;& t g 1 X t =0 p t & t subject to & t = F ( k t ;n t ) + (1 ¡ ¡ ) k t ¡ k t +1 ¡ w t n t ; 8 t ....
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This note was uploaded on 09/06/2011 for the course ECO 475 taught by Professor Hong during the Fall '07 term at Rochester.

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Homework8_sol - ECO 475 HW8 Solution Joon Song Assume the followings& u is strictly increasing and strictly concave& u is continuously

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