The Ramsey (Cass-Koopmans) Model October 2007 () Ramsey October - PowerPoint PPT Presentation

The Ramsey (Cass-Koopmans) Model October 2007 () Ramsey October 2007 1 / 16

Assumptions Neoclassical production function: Y t = F ( K t , A t L t ) where A 0 = L 0 = 1, and ˙ ˙ A t L t = g = n and A t L t Aggregate resource constraint: ˙ K t = Y t � C t L t � δ K t () Ramsey October 2007 2 / 16

Representative household’s problem Z ∞ e � ρ t C 1 � θ t max 1 � θ L t dt 0 subject to ˙ K t = r t K t + A t w t L t � C t L t and T ! ∞ D T K T � 0 . lim Each household has L t members and C t is consumption per household member. () Ramsey October 2007 3 / 16

The Intensive Form Aggregate resource constraint: ˙ k t = f ( k t ) � c t � ( n + g + δ ) k t . Competitive markets: f 0 ( k t ) r t + δ = f ( k t ) � k t f 0 ( k t ) . w t = () Ramsey October 2007 4 / 16

We can re-express the household’s utility as Z ∞ e � β t c 1 � θ t 1 � θ dt , 0 where the “e¤ective rate of time preference” is β = ρ � n � ( 1 � θ ) g . Dynamic budget constraint: ˙ k t = ˆ r t k t + w t � c t where the “growth adjusted” interest rate r t = r t � n � g ˆ () Ramsey October 2007 5 / 16

Transversality condition is ˆ D T k T � 0 . lim T ! ∞ where the “e¤ective discount factor” is D t = e � R t ˆ 0 ˆ r t dt . , ! intertemporal budget constraint: Z ∞ Z ∞ ˆ ˆ D t c t = k 0 + D t w t , 0 0 () Ramsey October 2007 6 / 16

Euler equation: c t ˙ r t � β ˆ = c t θ r t � n � g � ( ρ � n � ( 1 � θ ) g ) = θ r t � ρ � θ g = θ () Ramsey October 2007 7 / 16

Competitive Equilibrium Capital accumulation process ˙ k t = f ( k t ) � c t � ( n + g + δ ) k t Consumption growth = f 0 ( k t ) � δ � ρ � θ g c t ˙ c t θ Transversality ˆ lim D T k T = 0 T ! ∞ () Ramsey October 2007 8 / 16

. c(t) . k=0 k(t) Figure: The ( ˙ k = 0) locus () Ramsey October 2007 9 / 16 c t = f ( k t ) � ( n + g + δ ) k t

. c(t) c=0 k(t) k* c t = 0 ) locus Figure: The ( ˙ () Ramsey October 2007 10 / 16 f 0 ( k � ) = δ + ρ + θ g

. c(t) c=0 S . k=0 k(t) k* Figure: Phase Diagram () Ramsey October 2007 11 / 16

Balanced growth path Pair ( k � , c � ) where ˙ c = ˙ k = 0 : f 0 ( k � ) = ρ + δ + θ g c � f ( k � ) � ( n + g + δ ) k � . = Existence of BGP implies intersection to left of peak ! peak of ˙ k = 0 locus is at ˆ , k where f 0 ( ˆ k ) = n + δ + g ! k � < ˆ , k if ρ + δ + θ g > n + δ + g ρ � n � ( 1 � θ ) g > 0 e � β < 1 ) , ! necessary condition for utility to be …nite () Ramsey October 2007 12 / 16

Transitional Dynamics Capital, k t is a state variable Consumption c t is a jump variable Given k 0 , the intertemporal budget constraint determines c 0 . The unique path towards the BGP is the saddlepath trajectory () Ramsey October 2007 13 / 16

. c(t) c=0 S A . k=0 B D C k(t) k 0 k* Figure: Transitional Dynamics () Ramsey October 2007 14 / 16

Formal Derivation of Transitional Dynamics Log-linearization around the BGP We have the following system α k α � 1 � δ � ρ � θ g c t ˙ t = c t θ ˙ k t � c t k α � 1 = � ( n + g + δ ) t k t k t Let x t = ln k t and y t = ln c t , ! then α e ( α � 1 ) x t � δ � ρ � θ g y t = ˙ θ e ( α � 1 ) x t � e y t � x t � ( n + g + δ ) x t ˙ = () Ramsey October 2007 15 / 16

First-order approximation: α ( α � 1 ) e ( α � 1 ) x � [ x t � x � ] y t ˙ ' θ h ( α � 1 ) e ( α � 1 ) x � + e y � � x � i [ x t � x � ] � e y � � x � [ y t � y � ] x t ˙ ' Substituting in the steady state values: � ( 1 � α ) ( δ + ρ + θ g ) [ x t � x � ] y t ' ˙ θ � β + ( 1 � α )( n + g + δ ) � β [ x t � x � ] � [ y t � y � ] x t ˙ ' α () Ramsey October 2007 16 / 16