Ramsey Growth Model Dynamics

ECGA 7020 Macro Theory II Fall 2005

Fordham University Professor Darryl McLeod

Thanks to Rosendo Ramirez for the dynamic action slides

Consumption Laws of MotionConsumption Laws of Motion

• The key Euler equation determines the trajectory of consumption, the marginal rate of substitution between present and future is always equal to MRT between periods as determined by the production function, (the Keynes-Ramsey rule)

• Note that when f ’(k) is greater than + θg, c rises, and vice versa,

( ) ( ) (́ ( ))

( )

c t r t g f k t g

c t

Law of Motion for capitalLaw of Motion for capital

• The main innovation of the Ramsey model is to make savings s = y - c = f(k) – c endogenous, though we need a constant intemporal elasticity of substitution such as that provided by the CRRA utility function to do so.

( ) ( ) ( ) ( )k t sy t n g k t

( ) ( ( )) ( ) ( ) ( )k t f k t c t n g k t

y = f(k)

(n + g)k

Utility maximized when f ’(k) = ρ+θg

Utility ( c ) falls when

f’(k) < ρ+θg

kk*, f ’(k) = ρ+θg

Utility U(ct ) increases as long as f ’(k) > ρ + θg > n + g

Utility Maximization

y

ρ + θg

k

f ’(k)

c(t) increases

when

f ‘(k) > ρ + θg

when, f ‘(k*)=ρ+θg

c(t) falls when

f ‘(k) < ρ + θg

k*

.0c

f ’(k) = n+g

f ’(k) < ρ+θg

kk*

.0c

c

f ’(k) > ρ+θg

c > 0

c < 0

y = f(k)

(n+g)k

When f ’(k) = n+g, c reaches a maximum as

c = f(k) – (n + g)k (golden rule max c )

k

kgr (golden rule)

Dynamics of k: recall the golden rule: consumption reaches maximum when f’(k) = n +g

When f ’(k) > n+g, c increases.

When f ’(k) < n + g, c falls.

y

k

.0k ( ) ( ( )) ( ) ( )c t f k t n g k t

.0k

( ( )) ( ) ( ) ( )f k t n g k t c t

.0k ( ( )) ( ) ( ) ( )f k t n g k t c t

c

when

k falls in the green region because savings, s < (n+g)k .

when,

Savings, s > (n+g)k

Savings just covers investment per capita, s = y – c = (n+g)k

when,

Dynamics of k (the change in k depends on s = y-c )

k

cNote that k* < kgr (golden rule – see the next slide)

k* kgr (golden rule)

Note that golden rule kgr > k*

• The boundedness condition: we assume the discount rate ρ > 0 is large enough to assure that the present discounted value of utility is finite, that is,

β = ρ – n – (1-θ)g > 0. (see eq. 2.12 on page 52 of Romer (2001).

• And this β > 0 condition implies ρ+θg > n + g, so that k* < kgr.

k

c

k*

k = 0

c(t) = f(k(t)) - (n + g)k(t)

c(t) > f(k(t)) - (n + g)k(t)

k < 0

c > 0

(golden rule)k

c = 0

c < 0

c(t) < f(k(t)) - (n + g)k(t)k > 0

k

c

k*

k = 0

k < 0c > 0

(golden rule)k

c = 0

c < 0

c(t) < f(k(t)) - (n + g)k(t)k > 0

k

c

k*

k = 0

k < 0c > 0

(golden rule)k

c = 0c < 0

k > 0