Euler condition would no longer be satisfied. Likewise, if the initial capital level
0
k
was
to the right of
*
k
, there will be a unique stable manifold, satisfying the Euler and
Transversality conditions and the capital transition constraint, that will converge towards
the steady state
*
*
(
,
)
k
c
. If this was not to be the case, the economy would have ended up
with zero capital or zero consumption violating the physical or the Transversality
constraint. Summarizing results, we have established the following:
Proposition 1 (Neoclassical Growth Theory):
The Neoclasical Growth
Model economy, introduced in Section A, has a unique competitive equilibrium that
coincides with the solution to the Social Planner’s problem, (I.19).Along this equilibrium,
capital and consumption per efficient household satisfy conditions (I.35)(I.38), such that
the following are true:
(a)
There exists a unique (interior) steady state
*
*
(
,
)
k
c
, characterized by (I.41) and
(I.42).
(b) If
0
k
<
*
k
, {
{
,
}
1
1
0
k
c
t
t
t
increases monotonically as it converges towards
*
*
(
,
)
k
c
And, if
0
k
>
*
k
,
{
,
}
1
1
0
k
c
t
t
t
decreases monotonically as it
converges towards
*
*
(
,
)
k
c
Exercise 3
: Show that the competitive equilibrium of the Neoclassical Growth Model
satisfies the Kaldor Stylized Facts (i.e., output capital per capita increases, output per
17
capita increases, the capital – output ratio is constant, the return to capital is constant, the
capital and labor income shares are constant), along the steady state.
Exercise 4
: Show that the competitive equilibrium of the Neoclassical Growth Model
satisfies the Convergence hypothesis (i.e., the growth rate of less developed countries will
converge to the growth rate of developed countries monotonically).
E.
Local Stability Analysis
We can also characterize quantitatively the behavior of the competitive equilibrium
around the steady state. This is called local stability analysis. Here, it will be convenient
to work with the second order difference equation (I.34), directly. In fact, it is more
convenient to begin with the more general case of the Euler condition of the Basic
Problem, (I.27). If a steady state,
x
, exists, as in the case of the Neoclassical Growth
Model, this steady state must satisfy the following:
(
)
(
)
2
1
,
,
0
x x
x x
b
F
+
F
=
(I.43)
Moreover, if
F
is twice continuously differentiable, it follows by Taylor’s Theorem that
the LHS of (I.27) may be approximated, for
1
2
(
,
,
)
t
t
t
x
x
x
+
+
sufficiently close to
( , , )
x x x
,
as the LHS of the following:
(
)
(
)
(
)
(
)
(
)
(
)
2
1
21
22
1
11
1
12
2
,
,
,
(
)
,
(
)
,
(
)
,
(
)
0
t
t
t
t
x x
x x
x x
x
x
x x
x
x
x x
x
x
x x
x
x
b
b
+
+
+
F
+
F
+F

+F

+
F

+F

=
which, in view
of (I.43) simplifies to the homogeneous second order linear difference
equation with constant coefficients:
(
)
(
)
(
)
(
)
12
2
11
22
1
21
,
(
)
[
,
,
](
)
,
(
)
0
t
t
t
x x
x
x
x x
x x
x
x
x x
x
x
b
b
+
+
F

+
F
+F

+F

=
(I.44)
For the Neoclassical Growth Model, omitting function arguments, the coefficients of
(I.44) are given by:
1
2
n
z
2
11
12
n
z
2
22
n
z
Φ =u [f +(1 δ)] >0
Φ =u [(1)(1+ g )(1+ g )]<0
Φ =u [f +(1 δ)] +u f <0
Φ
=u [f +(1 δ)] [(1)(1+ g )(1+ g )]> 0
Φ
=u [(1+ g )(1+ g )]
<0
18
To solve (I.44) in a way that satisfies the initial and Transversality conditions, or