1 The Neoclassical Growth Model
The neoclassical growth model, which originated with the work of Solow
and Swan, consists of the following relationships: a production function,
y
t
=
f
(
h
t
;k
t
)
,wh
e
r
e
y
t
is output,
h
t
is labor, and
k
t
is capital at date
t
=0
;
1
;
2
:::
, plus a law of motion for the capital stock,
k
t
+1
=(1
¡
±
)
k
t
+
¾y
t
,
where
±
2
(0
;
1)
is the depreciation rate and
¾
2
(0
;
1)
is the savings rate.
We assume that
f
is homogeneous of degree 1, increasing, concave, and
twice continuously di¤erentiable. We assume for now that
h
t
is constant and
normalize
h
t
=1
(this assumption is not very accurate, but we will soon
endogenize
h
t
by letting agents choose how much time to spend working and
how much to spend in other activities). Let
F
(
k
)=
f
(1
;k
)
.T
h
e
n
F
0
>
0
,
F
00
<
0
. We further assume that
F
(0) = 0
,
F
0
(0) =
1
and
F
0
(
1
)=0
.Th
e
initial capital stock,
k
0
, is given exogenously.
Note that behavior is exogenous here: the representative individual in this
economy simply works a …xed number of hours
h
t
=1
,savesorinvests
i
t
=
¾y
t
, and consumes
c
t
=(1
¡
¾
)
y
t
, each period. Substituting the production
function into the law of motion for capital yields a …rst order di¤erence
equation in
k
t
,
k
t
+1
=(1
¡
±
)
k
t
+
¾F
(
k
t
)
´
g
(
k
t
)
:
(1)
Together with the initial condition
k
0
, (1) completely determines the entire
time path of the capital stock. Given this path we can compute the paths
for
y
t
,
c
t
,etc
.A
steady state
of the system is a solution to
k
=
g
(
k
)
.
1
1
One often sees the model in continuous time, in which case the law of motion for
capital is written
_
k
t
=
¾F
(
k
t
)
¡
±k
t
(see below for an explicit derivation of the continuous
time as the limit of discrete time models). Also, one often sees the model augmented
to include population growth, with the number of agents at
t
given by
N
t
=
e
nt
.I
n
this case, one needs to distinguish between total and per capital variables. Thus, let
Y
t
=
f
(
N
t
;K
t
)=
e
nt
F
(
K
t
=N
t
)
be the production function; then in per capita terms we
have
y
t
=
F
(
k
t
)
. Also, let
_
K