This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECO 475 HW8 Solution Joon Song Nov 11, 2009 Assume the followings. & u ( Â¡ ) is strictly increasing and strictly concave. & u is continuously di/erentiable and lim c & u ( c ) = 1 & F ( Â¡ ; Â¡ ) is strictly increasing, strictly concave and F ( k; 0) = F (0 ;n ) = 0 for any k Â¢ and n Â¢ . & 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 Â¡ 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 Â£ Â¢ ) k t Â£ k t +1 Â£ 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 Â¡ & ) k t ; 8 t . Characterization of ADE From consumer&s problem, we have u ( c t ) = Â¡u ( c t +1 ) p t p t +1 , 8 t . (1) From Â¡rm&s problem, we get p t p t +1 = F k ( k t +1 ;n t +1 ) + 1 Â¡ & , 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 ) = Â¡u ( c t +1 ) [ F k ( k t +1 ; 1) + 1 Â¡ & ] , 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 ) = Â¡u ( c t +1 ) [ F k ( k t +1 ; 1) + 1 Â¡ & ] , 8 t . (5) p t p t +1 = F k ( k t +1 ; 1) + 1 Â¡ & , 8 t . (6) w t = F n ( k t ; 1) , 8 t . (7) Â¢ t = F ( k t ; 1) + (1 Â¡ & ) k t Â¡ k t +1 Â¡ w t , 8 t . (8) n t = 1 , 8 t . (9) c t + k t +1 = F ( k t ; 1) + (1 Â¡ & ) 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 ;Â¢ 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 Â¢ 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 Â¡ 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 Â¡ Â¡ ) k t Â¡ k t +1 Â¡ w t n t ; 8 t ....
View
Full
Document
This note was uploaded on 09/06/2011 for the course ECO 475 taught by Professor Hong during the Fall '07 term at Rochester.
 Fall '07
 Hong

Click to edit the document details