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

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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:

SLIDE 1

The Ramsey (Cass-Koopmans) Model

October 2007

() Ramsey October 2007 1 / 16

SLIDE 2

Assumptions

Neoclassical production function: Yt = F(Kt, AtLt) where A0 = L0 = 1, and ˙ At At = g and ˙ Lt Lt = n Aggregate resource constraint: ˙ Kt = Yt CtLt δKt

() Ramsey October 2007 2 / 16

SLIDE 3

Representative household’s problem

max

Z ∞

eρt C 1θ

t

1 θ Ltdt subject to ˙ Kt = rtKt + AtwtLt CtLt and lim

T !∞ DT KT 0.

Each household has Lt members and Ct is consumption per household member.

() Ramsey October 2007 3 / 16

SLIDE 4

The Intensive Form

Aggregate resource constraint: ˙ kt = f (kt) ct (n + g + δ)kt. Competitive markets: rt + δ = f 0(kt) wt = f (kt) ktf 0(kt).

() Ramsey October 2007 4 / 16

SLIDE 5

We can re-express the household’s utility as

Z ∞

eβt c1θ

t

1 θ dt, where the “e¤ective rate of time preference” is β = ρ n (1 θ)g. Dynamic budget constraint: ˙ kt = ˆ rtkt + wt ct where the “growth adjusted” interest rate ˆ rt = rt n g

() Ramsey October 2007 5 / 16

SLIDE 6

Transversality condition is lim

T !∞

ˆ DT kT 0. where the “e¤ective discount factor” is ˆ Dt = e R t

0 ˆ

rtdt.

, ! intertemporal budget constraint:

Z ∞

ˆ Dtct = k0 +

Z ∞

ˆ Dtwt,

() Ramsey October 2007 6 / 16

SLIDE 7

Euler equation: ˙ ct ct = ˆ rt β θ = rt n g (ρ n (1 θ)g) θ = rt ρ θg θ

() Ramsey October 2007 7 / 16

SLIDE 8

Competitive Equilibrium

Capital accumulation process ˙ kt = f (kt) ct (n + g + δ)kt Consumption growth ˙ ct ct = f 0(kt) δ ρ θg θ Transversality lim

T !∞

ˆ DT kT = 0

() Ramsey October 2007 8 / 16

SLIDE 9

c(t) k(t) k=0 . . Figure: The ( ˙ k = 0) locus

ct = f (kt) (n + g + δ)kt

() Ramsey October 2007 9 / 16

SLIDE 10

c(t) k(t) c=0 k* . Figure: The (˙ ct = 0) locus

f 0(k) = δ + ρ + θg

() Ramsey October 2007 10 / 16

SLIDE 11

S c(t) k(t) c=0 k=0 k* . . Figure: Phase Diagram

() Ramsey October 2007 11 / 16

SLIDE 12

Balanced growth path

Pair (k, c) where ˙ c = ˙ k = 0 : f 0(k) = ρ + δ + θg 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 + θg > n + δ + g ρ n (1 θ)g > ) eβ < 1 , ! necessary condition for utility to be …nite

() Ramsey October 2007 12 / 16

SLIDE 13

Transitional Dynamics

Capital, kt is a state variable Consumption ct is a jump variable Given k0, the intertemporal budget constraint determines c0. The unique path towards the BGP is the saddlepath trajectory

() Ramsey October 2007 13 / 16

SLIDE 14

S c(t) k(t) c=0 k=0 k* k0 C D B A . . Figure: Transitional Dynamics

() Ramsey October 2007 14 / 16

SLIDE 15

Formal Derivation of Transitional Dynamics

Log-linearization around the BGP

We have the following system ˙ ct ct = αkα1

t

δ ρ θg θ ˙ kt kt = kα1

t

ct kt (n + g + δ) Let xt = ln kt and yt = ln ct , ! then ˙ yt = αe(α1)xt δ ρ θg θ ˙ xt = e(α1)xt eytxt (n + g + δ)

() Ramsey October 2007 15 / 16

SLIDE 16

First-order approximation: ˙ yt ' α(α 1) θ e(α1)x [xt x] ˙ xt ' h (α 1)e(α1)x + ey x i [xt x] ey x [yt y ] Substituting in the steady state values: ˙ yt ' (1 α) (δ + ρ + θg) θ [xt x] ˙ xt ' β [xt x] β + (1 α)(n + g + δ) α

[yt y ]

() Ramsey October 2007 16 / 16