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ON STABILITY IN THE SADDLE-POINT SENSE
David Levhari and Nissan Liviatan
Number 73
July 1971
massachusetts
institute of
technology
'memorial drive
Cambridge, mass. 02139
Dewey
ON STABILITY IN THE SADDLE-POINT SENSE
David Levhari and Nissan Liviatan
Number 73
July 1971
This research was supported by National Science Foundations grants
GS-2530 for the first author and GS-1812 for the second author.
The
views expressed in this paper are the authors' sole responsibility and
do not reflect those of the Department of Economics, Massachusetts
Institute of Technology or the National Science Foundation.
175.1060
.
OCNCHAl. OOOKDINOING CO
^•^
5551
E3
13
«U«LlTr CONtKOL MARK
Introcuction
In the following paper we discuss some properties of the op-
path
(..•i<..il
the multiple capital goods case.
Lit.
We denote by
n-dimensional veciOr of per capita capital and by
the
^(k,k)dt
fe
V
where
the
We ass;ame that the aim is to max-
rates of change par unit of time.
imize
k
k
gives the maximum utility obtained
i'
from c:onsumption with given k and k, and
6
is the rate of dis-
count.
This is the case discussed in numerous papers and here
we focus
our attention on the behavior of the Euler differential
equations around the steady state.
Samuelson
[5]
has proved that for the case
6=0
we have, at
an optimal steady state, a saddle-point with the characteristic
roots coming in pairs of
and
X
~X
For the analogues discrete
.
model he has proved [6,7] that the roots come in reciprocals
and
?.
X
Thus, in both cases the behavior of the path around the
steady state is that of a saddle-point.
We refer to this situation
as "stability in the saddle-point sense".
Kurs
[1]
has generalised this Samuel son-Poincare
considering the case of
count rate.
<S
>
0,
theorem by
that is the case of positive dis-
Kurz proves for this case that it is impossible for
the optimal path to be^'^^table one; either we have a saddle-point
around the steady state ofV^IHstability.
* We T7ish to acknowledge NSF grants GS-253
f or
Thus an optimal steady
the first author and GS-1812
for the second author.
1)
.
.
See Similar formulation in Samuelson and Solow
[4]
- 2 -
state may be stable or unstable xn the saddle-point sense.
2)
Kurz's presentation is different from that of Samuelson since his
treatnent is based on a generalized system which includes the shadow
prices. as
wel^i..
cand con s iderably simpler^
>^^/
In this paper we give^lternativ^V^roof to the Kurz theorem
within the framework of classical calculus of variations, as
Samue.-son is using, without introducing dual variables.
for
;)
that if
>
X
is a characteristic root then
also a characteristic root.
We prove
-X+6
is
'
Another problem which we take up in this paper concerns the
possibility of having purely imaginary roots
6=0.
and
X
for the case
If this were possible then it would imply lack of saddle-
point stability for an optimal steady state with
seems very strange.
[5,7]
-X
Indeed
that for the case
<,
6=0
6=0, which
Samuelson 's writings seem to imply
and concave production function and
utility function (that is strictly concave
V)
it is impossible
for the characteristic roots to be purely imaginary.
That is,
his conjecture is that when the integrand is strictly concave in
e
k
and
k
it is always the case that the optimal path has a
saddie-point behavior.
Kurz, on the other hand, raised the pos-
sibi.lity of total instability for the case
6=0.
Kurz does not
state, however, whether he is discussing the case of a concave in-
tegrand or not.
that when
2)
6=0
In this paper, we prove as Samuelson conjectures,
the optimal path around the steady state has in-
Tne case of total instability is connected with multiple optimal
steady state in which some are saddle-points and some are unstable
[2,3].
~ 3 -
deed the saddle-point behavior for strictly concave
V.
All the theorems are proved also for the discrete time model.
Zr.i
this case,, the aim is to maximize
v(k^k,
3
j;
where
,)
k
t=0
n-vect-or of per capita capitals at
is
(0
<
g
3
V
and
1)
is the discount factor
3
arguments function giving the max-
2n
is a
t,
imum utility of consumption for known
We prove
and
k
,
that the characteristic roots come in pairs of
X
and
the case
V
and
3=1
^
k
.
^r
.
In
strictly concave it is impossible for any
of the roots to be of absolute value
and we always have a
1
This is the discrete time counterpart of the theo-
saddle-point.
rem stating that in continuous-time models we cannot have purely
imaginary roots when
2.
6=0
strictly concave.
V
and
The Continuous and Discrete Models.
In the continuous case we wish to maximize
As shown in
[4]
and
(V^.^.)(y) +
=
l,...,n
f
j^
)-6(V3^
j^
13
31
)]
-
y
[(V,,.k.)+6Vj^
13
j^
](y)
13
n-vectors of deviations from optimal steady
are
state values and where
i,j
)-(V3^
3^
13
y = k-k*
whera
yields
(k=0)
HVj^
13
.
the Taylor's expansion o± Euler Equations
[5]
around the steady state
(1)
\;'(k,k)dt
fe
(V^ ^ )
k^kj
,
(V^
,
k^kj
)
,
(V,
^
k^kj
,
(V,
,
k^kj
)
,
with
matrices composed of
nxn
are notations for the
)
the appropriate partial derivatives evaluated at an optimal steady
state.
Denote
A =
(Vj^
^
i
observe
B' =
(V^
,
3
)
)
j
^
S =
^^k
)
j^
i
j
^
^ "
^\ i k
^
'
^^'^
J
(prime denotes the transposed matrix)
2.
characteristic equation of this system iss
"®
.
The
=
(2)
'
(C+dB^I! =
(B-B'-(SA)X -
+
IJAX^
.
A
Under the assumption of strict concavity the matrix
(_
B
,
B
_)
C
is negative definite,
For the discrete case, as mentioned, the aim is to maximize
00
Z
t
3
V(k
,k.
V
where
,)
gives max
itial
utility for given
k. ,k.
,,
Taylor expansion of the system of Eu ^er-like difference equations
3)
1^
yields:
•
3(V
(3)
.)(^'t+2^
^
^^\i^j>-^^^V
Vt
^t-l^t
+ (V
.
values, i.e.,
x.
V
=0
.
{V ^
t— i
-I
A = h
= B'«
.
kj
= k.-k*.
W
We denote
observe
)
n-vectors of deviations from optimal steady state
are
x.
(x.
V'^
Ic
where
)
.
)5(^t^l^-^
1.3
^t-l^t-1
,
{V ^
.
)
= B
W~i
,
(V ^
.
)=C
and
^t-i^t-1
The characteristic equation of the Euler-
K.'j_
c.
like system of difference equations is
iJgB'X^ +
(4)
CA+3C)A +
B|J
=
.
Characterizations of the Optimal Path
Theorem
tion
(2)
Proof.
1.
If
then
X
-X+6
is a characteristic root of the polynomial equais a root as well.
Let us substitute
A{6-X)^ + (B-B'-6A)(6-|^)
3)
-X+6.
- (C+6B}
This yields with trivial calculations
= AX^
-S-
(B-B •+6A) (-X)
-
(C+6B')
See the analogous case in Samuelson [5,7] with characteristic
equation of the same form.
.
:
'
- 5 -
However, we already know that
|jAX^
'
+
(B-B''-6a)X -
(C+6By
=
||
and
as transposing does not change the value of the determinant, we
This theorem gives Kurz's con-
is also a root.
-X+6
find that
clusions that the only possibilities are either saddle-point or
complete instability.
We cannot have complete stability since if,
say, the real part of
X
related root
is negative then the real part of the
cannot be negative too.
-X+&
In the following theorem, we show that in the case
6=0
we
always have a saddle-point, that is purely imaginary characterixtic
roots are impossible.
Theorem
The characteristic equation
2
'
|}AX
+
(B-B')X -
posesses
i\\
no purely imaginary root.
Assume
Proof.
AX
2
(AX
= 10
(B-B')X - C
-s-
2
X
to be a characteristic root.
possesses a non-trivial solution
{B-B')X - C) (x+iy) =
+
The matrix
real vectors not both zero.
where
x
and
y
are
n-dimensional
Multiplying this equation by
(x-iy)
we get
(5)
(x-iy)
'
[AX^ +
(B-B')X - C] (x+iy) =
.
Let us now compute the following quadratic form (defining
to be the conjugate of
(6)
[X
(x-iy)
',
(x-iy)
X)
'I
+ X (x-iy) 'B (x+iy)
As
(7)
X
X
(^
^')
L^^+iy^^
+ X (x-iy) 'B' (x+iy)
X
is purely imaginary
-(x-iy) 'A (x+iy) X^ + (x-iy)
Thus, we see that:
'
= -X
= XX(x-iy) 'A(x+iy)
^
+
(x-iy) 'C (x+iy)
and we obtain for
(B'-B) (x+iy) X +
+
.
(6)
(x-iy) 'C (x+iy)
.
.
.
'
- 6 -
[X(x-iy);x-iy]' (^
(8)
=
^')
[^^^^J^^]
= -(x-iy)'[AX^ + (B-B')X - C] (x+iy) =
where the equality to zero follows from
A
However, the matrix
B
_
(»
(5)
is a symmetric negative definite
)
matrix and for any negative definite symmetric matrix
quadratic for
z'Hz",
left hand side of
as in
(8)
is negative for
(5),
z
H, the
0.
^^
Hence, the
should be negative, a contradiction.
It
follows that all the roots of this characteristic equation must
have non-vanishing real parts.
Theorem
(4)
then
Proof.
(The discrete case)
3
1
yq
If
,
is a solution of equation
X
^^ also a solution.
By substitution in
&B'
(^)2
+
we obtains
(4)
(A+BC)^
Multiplying all the elements by
trix is multiplied by
B^X^^
Transposing, we finally have
B
-}-
BX
.
2
the determinant of this ma-
and we get
BB'X
already know that ^|(BB'X^ + (A+BC)
2
+
(A+BC)
+
X
+
B'
(A+BC)
= 0.
^|j
+ BBX^.
However, we
+ B.
X
X
Thus,
^
is also
a root.*
We find then that it is impossible for both roots to be within
the unit circle.
In the case
1 <
[Xj
<
^
both roots will be
outside the unit circle and we shall obtain the unstable case.
the case of no discounting,
B=l
non-saddle-point is the case of
Ail this assumes that
singular case.
X
j^
t
In
the only conceivable case of
|XJ
=1.
or that
^
However, the following
j^H
f^
i
we neglect this
- 7 -
theorem shows that in the strictly concave case this is impossible.
Theorem
The equation
4.
+
^|jB'X
(A+C) X +
b(|
=
possesses no
roots on the unit circle.
Assume that this equation possesses a root on the unit circle,
i.e.,
X
As
is a root with
B'X
2
+
X-X
=1.
is a singular matrix, there is a non-
(A+C) X + B
trivial solution to the system.
(A+C)X + B] (x+iy) =
[B'X^ +
Multiplying on the left side with
X(x-iy)
.
and using
°
XX =
we
1
find
(9.!
(x-iy)'[B'X + (A+C) + BX] (x+iy) =
.
Let us look at the following quadratic form
(10)
[X(x-iy)',(x-iy)']
However, as
(A+C)
A
(
B
,
^)
=
[^(^tiy)]
(x-iy)
'
(A+C) (x+iy)
+
(x-iy) 'B' (x+iy) X =
(x-iy) 'B(x+iy)X +
(x-iy) '[B'X +
(^,
)
+ bX] (x+iy)
=
.
...
.
is negative definite symmetric matrix
.
multiplied on both sides by conjugate vectors it should be negative, a contradiction.
3=1
Thus, also for this case, we find that
is always a case of saddle-point and again, complete instabi-
lity is impossible in the case where
V
is strictly concave.
REFERENCES
[1]
Kurz, M.
,
"The General Instability of a Class of Competitive
Growth Processes'; Rev. Economic Studies 35 ^ 1968, 155-174.
[2]
,
'Optimal Economic Growth and Wealth Effects'^ Int.
Economic Rev.
[3]
9
[1968], 348-357.
Liviatan, N. and P. A. Samuel son, "Notes of Turnpikes: Stable
and Unstable" J. of Economic Theory
,
Vol.1,
Dec, 1969,
454-475.
[4]
Complete Capital Model Involving Heterogenous Capital Goods'^ Quarterly J. of Economics,
Samuelson, P. A., Solow, R.M.
,
"A
1956, Vol.70, 537-567.
[5]
Samuelson, P. A., "Efficient Paths of Capital Accumulation in
Terms of the Calculus of Variations", in Math. Studies in
the Social Sciences
,
1959, editors: Arrow, K.J., Kalin, S.,
Sappes, P., Stanford, 1960.
[6]
[7]
"Reciprocal Characteristic Root Property of
Discrete Time Maxima", Western Economic J. 6, 1968, 90-93.
,
,
"Local Proof of the Turnpike Theorem", West-
ern Economic J. 7, 1969, pp. 1-8.
Date Due
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