Dynamic Macroeconomic Modeling with Matlab

4 Numerical Simulation of Macroeconomic Models
4.2 The Ramsey Model
4.2.1 The model (social planner’s solution)
Consider a simple neoclassical economy without technological progress. The production func-
tion is neoclassical with labor and capital as inputs according to
or
Y = K α L 1−α
y = k α
0 < α < 1. (21)
in per capita terms. The utility function of the representative individual is of the CIES type
u(c) = c1−σ
1 − σ
The economies resource constraint is given by
with depreciation δ and population growth rate n.
A social planner solves
(22)
with σ > 0. (23)
˙k = y − c − (n + δ)k (24)
∞
c
max
c(t) 0
1−σ
1 − σ e(n−ρ)tdt (25)
s.t. ˙ k = y − c − (n + δ)k k(0) = k0
The current-value Hamiltonian of this problem reads
and the necessary first-order conditions are given by
H = c1−σ
1 − σ + λ(kα − c − (n + δ)k) (26)
Hc = 0 ⇔ c −σ = λ (27)
Hλ = ˙ k ⇔ ˙ k = y − c − (n + δ)k (28)
Hk + (n − ρ)λ = − ˙ λ ⇔ − ˙ λ = (αk α−1 − (n + δ))λ + (n − ρ)λ (29)
and the transversality condition
lim k(t)λ(t) = 0. (30)
t→∞