Discussion Paper No.340
Delay Solow Model with a CES Production Function

Akio Matsumoto
Chuo University

Ferenc Szidarovszky
Corvinus University

January 2021

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanDelay Solow Model with a Normalized CES
Production Function
Akio Matsumotoy

Ferenc Szidarovszkyz

Abstract
This study is aimed at analyzing the relationship between the elasticity
of substitution and the birth of cyclic macro-dynamics. We …rst normalizea CES production function that consistently involves all its variants. Introducing two di¤erent delays, called a gestation delay and a depreciation
delay, into the Solow model, we then …nd that the gestation delay alone
does not a¤ect the stability of the balanced growth equilibrium. Third,
we construct the stability switching curve that separates the stable andunstable regions in the plane of the two delays and derive the stability
index to con…ne whether the stability is lost or gained when the two delays cross the curve. Lastly, we numerically demonstrate that appropriate
combinations of two delays can be a source of oscillatory ‡uctuations.

Keywords: Normalized CES production function, Solow model, Gestation delay, Depreciation delay, Stability switching curve

The …rst author highly acknowledges the …nancial supports from the Japan Society for
the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 20K01566). The usual
disclaimers apply.
y Professor, Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji,
Tokyo, 192-0393, Japan. akiom@tamacc.chuo-u.ac.jp
z Professor, Department of Mathematics, Corvinus University, Budapest, Fövám tér 8,1093, Hungary. szidarka@gmail.com

1

1

Introduction

In the recent literature, it is demonstrated that traditional growth models might
be cyclic if time delays are introduced. See Tsuzuki (2016) for a new Keynesianmodel, Gori et al. (2017) for a Keynesian model, Guerri et al. (2019) andMatsumoto and Szidarovszky (2020) for a Solow model and an extended Solowmodel augmented with human capital. This …nding indicates that delay growthmodels could explain the cyclic behavior of various economic variables. On theother hand, in the existing growth literature, there has been a rising interest on
the relation between factor substitution and economic growth. In consequence,in view of a systematic comparison between production technologies and various
elasticity of factor substitution, a normalized CES production function has beendeveloped (see for example Klump and Preissler (2000)). Nevertheless, little is
known about dynamic behavior of a delay Solow model having a CES production
function. This study investigates the conditions under which the delay Solowmodel gives rise to oscillatory ‡uctuations. The model is delayed because it
involves a gestation delay of investment and a depreciation delay of capitalstock. Beside, it adopts a normalized CES production function.
The Solow model with no delays concerns the long-run evolution of the economy. After the economy reaches its steady state, it describes from a more longrun point of view how per-capita output grows via exogenous factors changes
such as population growth, technological progress and the saving rate. On theother hand, oscillatory ‡uctuations that are the main concern of macroeconomics are often observed in a real economy. In his classic development of thetheory of business cycle, Kalecki (1935) discovers that a delay of investment
called a gestation delay could be the main source of macro-cyclic dynamics. In
the early 50s, Hicks (1950) and Goodwin (1951) credit macro-cycles with nonlinearities.1 In later periods, Day (1982) gets down to the nitty-gritty of nonlinearity and shows that the Solow model with nonlinear production function
depending on the in‡uence of pollution can generate complex dynamics involving chaos in a discrete-time framework. Matsumoto and Szidarovszky (2011)
reconsider Day’s discrete time model in a continuous-time framework and show
the birth of chaotic dynamics if the delay in investment becomes strong enough.
It conveys a new result that nonlinearities and the delay could be responsible
for the birth of chaotic dynamics in continuous-time models.
This study is a continuation of Guerrini et al. (2019) that consider the samemodel with a Cobb-Douglas production function and determine the stability
switching curve on which stability is lost or gained. Its main purpose is tomake clear the relation between the elasticity of factor substitution and thestability-loss-and-gain. The main results are the followings:(i) Complex dynamics can emerge more likely if the elasticity of substitution
is smaller;
1 Goodwin’s model (1951) has a delay due to the acceleration principal, however, his main
dynamic equation is linearized at the zero-delay point and thus gets rid of the in‡uence caused
by delay. The existence of a limit cycle is mainly due to the nonlinearities of the investment
function.

2

(ii) Multiple stability switches can occur if the elasticity of substitution is larger
than unity.
The rest of the paper is organized as follows. Section 2 constructs a normalized CES production function that includes all variants. Section 3 reviews the
Solow model with no-delays and examines the dynamic properties of the Solowmodel with one delay. Section 4 extends the model by introducing two di¤erent
delays and analytically derives the stability switching curve. Finally, concluding
remarks and further research directions are given in Section 5.

2

Normalized CES Production Function

Let Y = F (K; L) be a production function having two inputs, capital K andlabor L; FK and FL are the marginal products of capital and labor, FKL is
the cross derivative. The two de…nitions of the elasticity of substitution between capital and labor were given, independently, by Hicks (1932, p.244) andRobinson (1933, p.256). The formulation de…ned by Hicks (1932) is as follows:
H =

FK FL
:
FKL Y

(1)

The economic implication of this de…nition is not clear. The other formulation
by Robinson (1933) isd log (K=L)(2)
R =d log (FL =FK )where the elasticity of substitution is de…ned as the relative change in thecapital-labor ratio against a one percent change in the marginal rate of technicalsubstitution. It has been well-known that these de…nitions are equivalent if the
production function has only two input factors and is homogeneous of degreeone, furthermore, the market is perfectly competitive.2 Under the degree-one
homogeneity (i.e., f (k) = F (K=L; 1) with k = K=L), the elasticity of substitution (2)3 can be written as=

f 0 (k) [f (k) kf 0 (k)]kf (k)f 00 (k)(3)

where f 0 (k) and f 00 (k) are the …rst and the second derivative of f (k).
The form of the corresponding production function is shown by Arrow etal. (1961). In particular, integration of (3) leads to an aggregate production
function in an extensive form,
1

Y =

(4)

L + K

2 A formal and compact proof shown by Hicks (1963, p.373) is summarized in the Appendix.
3 Since the equivalence has been shown, we denote the elasticity of substitution by3:

where and are some arbitrary integration constants and the parameter is
a proxy of ;
1
=
for 0
1
(5)
implying that strictly increases in and 1
1. Notice that (4) is asolution of (3), the second-order di¤erential equation in k, and has dependency
on the initial point. As is fully emphasized by Klump et al. (2012), the elasticity
of substitution is de…ned as a point elasticity and as a natural consequence, theintegration constants of (4) could depend on the baseline values. Hence, thenormalized CES function depends on three given baseline values:
K0: capital intensity,
L0

k0 =
0 =

FL
FK

: the marginal rate of technical substitution
0:K=K0 ;L=L0

and
y0 =

Y0
: per-capita output.
L0

Denoting the right-hand side of (4) by F (K; L); we have the marginal rateevaluated at the baseline,
FLFK

0:K=K0 ;L=L0

from which
=

=

0

1

K0
L0

0

1

L0
K0

=

.

(6)

This
is substituted into (4) and evaluated at the baseline that has, after
arranging the terms, the following form,
Y0 =Solving this equation for

h

K0

1

(K0 +

presents
=

i1
:
L
)
0
0

Y0
K0

1

(K0 +

0 L0 ):(7)

Substituting (6) and (7) into (4) and, again, arranging the terms yield
Y = Y0where"

K0
K0 + 0 L0

K
K0

+

0 L0

K0 +

0 L0

L
L0

(FK )0 K0
K0
r0 K 0==K0 + 0 L0(FK )0 K0 + (FL )0 L0
Y0
4

#1

and

0 L0

K0 +

0 L0

(FL )0 L0
w0 L0=:(FK )0 K0 + (FL )0 L0Y0=Here r0 and w0 are the rate of return and the real wage rate, both having output
as numéraire and satisfy the pro…t maximizing conditions,r0 = (FK )0 and w0 = (FL )0 :Let 0 be the capital share in the total income at the point of normalization,
then
r0 K 0
w0 L0
and 1
:
0 =
0 =
Y0
Y0
Hence, the normalized CES production function has the following form:
Y = F (K; L) = Y0

"

K
K0

0

+ (1

L
L0

0)

#1

:

(8)

It is well-known that the CES function includes the three most known forms of
production technologies as its special cases: The linear production technology
when ! 1 (i.e., ! 1);
Y = Y0

0

K
K0

+ (1

the Cobb-Douglas technology when

! 0 (i.e.,0K
K0

Y = Y0and the Leontief technology when

!

L
L0

0)

L
L0

;

! 1);
1

1 (i.e.,

0

;
! 0).

K L
;
:
K0 L0

Y = Y0 min

Under the linear homogeneity of the function F (Kt ; Lt ), the corresponding intensive form of (8) is

y = f (k) = y0
where
y=

"

0

k
k0

+ (1

Y
K
; k=
and f (k) = F
L
L

5

#1

0)

K
;1 :
L

(9)

The intensive forms in the special cases arey = y00y = y0kk0

y = y0 min

k
k0

+ (1

0)

when

! 0;

when

! 1;

0

k
;1
k0

when

!

1:

We will take the following speci…cation of the baseline whenever we give numerical examples. Under Assumption 1, Figure 1(A) depicts the three di¤erent
shapes of the isoquants, F (K; L) = Y0 ; with the di¤erent values of while Figure 1(B) represents the graphs of the corresponding intensive forms. It is seen
in both …gures that the curves pass the common baseline values denoted by theblack points and are tangent to each other there.
Assumption 1. Y0 = 1; K0 = 1; L0 = 2 and r0 = 1=3:(A) Extensive forms(B) Intensive forms

Figure 1. Special cases of the normalized CES production function

3

Delay Solow Model

The fundamental equation of capital accumulation in the Solow model is described by a one-dimensional di¤erential equation,_k(t)= sf (k(t))

6(n + )k(t)(10)

where k(t) is the per-capita capital at time t, f (k) is the intensive form of
the CES production function given in (9), s 2 (0; 1) the constant marginal
propensity to consume, 2 (0; 1) the depreciation rate of the capital and n 2(0; 1) the growth rate of population (that is, employment in the neoclassicalframework). We introduce delays in this dynamic equation to see how thedelays a¤ect the growth of the economy. To this end, we adopt the delay capital
accumulation equation formulated by Guerrini et al (2019) in which the CobbDouglas production function is used,_k(t)= sf (k(t(n + )k(t
(11)
1 ))
2)
where 1 is called a gestation delay and 2 a depreciation delay.4 This section
is divided into three subsections and consider the following cases: the no-delaycase (i.e., 1 = 2 = 0) as a benchmark, the one-delay case (i.e., 1 > 0 and
5
2 = 0) and …nally the equal-delay case (i.e., 1 = 2 = ). The case of twodi¤erent delays will be studied in the next section.

3.1

No Delays

The fundamental capital accumulation dynamics without delays is described by_(10). For simplicity, let be the sum of n and . A steady state with k(t)=0
is reached when the capital intensity k solves the following:
f (k)
=
k
s
or

0

B
k =@

1

(12)
11

0

k0
y0 s

0

C
A :

(13)

_
As is well-known, k(t)= 0 implies that at the steady state, all of the investment
is used to complement depreciation of the existing capital stock and additional
capital for the incoming population.
Before proceeding we con…rm how the equilibrium capital depends on theparameter , which measures the elasticity of substitution between L and K.
The average product of capital, the left-hand side of equation (12), is rewritten
as
"
#1

f (k)
1
0
0
= y0
+
k
k
(k0 )
and its derivative is negative in k for all values of ;
d
dk

f (k)
k

=

(1

f (k)=k

0)
0

k
k0

+ (1

< 0:
0)

4 See Guerrini et al. (2019) for justi…cation for the di¤erent length of the depreciation delay
from that of the gestation delay.
5 Since the other one-delay case of
1 = 0 and 2 > 0 is unrealistic from the economic pointof view, we omit it.

7Consider, …rst, the case of high elasticity of substitution with 0 <then turn to the case of low elasticity of substitution with < 0.6
Case I: 0 << 1 and<1

The limits of the average product are
lim g(k; ) = ( ) > 0

k!1

with
g(k; ) =

1
f (k)
y0
and ( ) =( 0) > 0kk0

andlim g(k; ) = 1:

k!0

Given , the average product is downward-sloping and is asymptotic to thepositive constant, ( ). If =s is large enough so thats

> ( );(14)then equation (12) has a solution k ; as seen in Figure 2. Given the parametervalues as A = 1=2; 0 = 1=3; y0 = k0 = 1=2; the graph of the average productg(k) against positive k is illustrated as the downward sloping curve in the …rst
quadrant and the graph of ( ) against positive is depicted as the positive
sloping curve in the second quadrant.7 Notice that
(1) =

y0 0
and lim ( ) = 0:!0
k0

If the direction of the inequality in (14) is reversed, then g(k) > =s holds_
for only k
0 and the sustainable growth (i.e., k(t)> 0) takes place. It isapparent from Figure 2 that a stronger su¢ cient condition for the existence ofthe equilibrium is> (1)sunder which the equilibrium capital always exists in the high elasticity of substitution case.
6 See Klump and Preissler (2000) for more detail.
7 Please observe that we cut and paste two di¤erent …gures to construct Figure 2.

8

Figure 2. Determination of k : 0 <Case II:<1<0

The average product approaches zero as k increases in…nitely and converges
to some positive constant as k goes to zero,lim g(k; ) = 0

k!1

and
lim g(k; ) = ( ) > 0:

k!0

If =s is small enough so that
< ( );(15)sthen equation (12) has the solution as seen in Figure 3 in which the graph of g(k)is illustrated in the …rst quadrant as in Figure 2 while the graph of ( ) against
negative is depicted as a downward sloping curve in the second quadrant. If
condition (15) is violated, then g(k)=s < 0 for all k and the capital stock
declines continuously. Notice thatlim ( ) = 1 and
!0

lim

! 1

( )=

y0
:
k0

A stronger su¢ cient condition for the existence of the equilibrium capital stock
is
y0
<
s
k0

9

under which the equilibrium capital always exists in the low elasticity of substitution case.

Figure 3. Determination of k :Since f (k)=k strictly decreases, f 0 (k)kf 0 (k ) <Therefore if<0

f (k) < 0 so at the steady state,f (k )= :ksdenote the capital share at the steady state, then=f 0 (k )kk<= 1:
f (k )
sf (k )To check the local stability at the equilibrium, we linearize the accumulation
equation in the neighborhood of the equilibrium_k(t)= [sf 0 (k )] k(t)(16)

where
sf 0 (k )< 0:Hence the eigenvalue of the linearized equation is negative. Summarizing theresult yields the followings:
Theorem 1 If either =s > ( ) for 0 < < 1 or =s < ( ) forthe equilibrium k > 0 exists and is locally asymptotically stable.

10< 0, then3.2

One Delay

Suppose that there is only an gestation delay, > 0; of investment in the capital
accumulation, which Kalecki (1935) considers for his macroeconomic dynamic
study,_k(t)= sf (k(t
))
k(t)
(17)
where k in (13) is also a stationary point. The linearized system is
_
k(t)
=

k(t

)

k(t)

where
sf 0 (k ) =The corresponding characteristic equation ise+ = 0:(18)

Assume that = i! with ! > 0. The real and imaginary parts of the characteristic equation arecos ! =
andsin ! =!:

Squaring both sides of these equations and adding them give
!2 =

2

(

+ 1) (

1) < 0:Hence there is no ! > 0 satisfying the characteristic equation (18). No stability change to instability from stability occurs. This result has been alreadycon…rmed in the case of a Cobb-Douglas production function (see Guerrini etat. (2019), Lemma 2). We con…rm it under the normalized CES production
function, implying that the gestation delay is harmless to the stability of thebalanced growth equilibrium.
Theorem 2 The equilibrium point k of the one delay capital accumulation
equation (17) is locally asymptotically stable for any > 0.

3.3

Equal-Delays

Since
as

1 =

2 =

is assumed, the delay dynamic equation (11) can be written_k(t)= sf (k(t

))

k(t

)

(19)

where, again, k in (13), is a stationary point. The equal-delays means that thecapital depreciation starts when the capital becomes productive. The linearized
equation is obtained as
_
k(t)
=

(1

) k(t
11

):

Suppose this equation has an exponential solution,k(t) = e t u:
The corresponding characteristic equation is written as
+ (1

) e

= 0:

(20)

Substituting the exponential solution into the characteristic equation (20) andseparating the real and imaginary parts yield two equations,(1) cos ! = 0

and
!

(1

) sin ! = 0:

Squaring both sides of these equations and adding them give2

! 2 = (1The critical values
`

)

2

> 0:

that satisfy these two equations are

( )=

h

1
(1

( ))

2

i

+ 2`

for ` = 0; 1; 2; :::(21)

The direction of stability switches can be determined as follows. Consider
as a function of , the bifurcation parameter, and di¤erentiate the characteristic
equation (20) with respect to ;
0

+ (1

0

) e

=0

implying that
2
0

=

1+
where the characteristic equation is used to derive this form. If
0== i!; then

!2 i !3
1 + 2 !2with a positive real part. So at the smallest critical value 0 ( ), stability is lost,and it cannot be regained with larger values of . Hence, the results obtainedin the case of the equal delays are summarized as follows:
Theorem 3 The equilibrium point k of the equal-two-delay equation (19) exists and is locally asymptotically stable for < 0 ( ); loses stability at = 0 ( )and bifurcates to a limit cycle for > 0 ( ).

12Under Assumption 1 with
= 1=10; s = 1=5 and ` = 0, the stability
switching curve against is obtained as a positive sloping curve as illustrated
in Figure 4(A). The steady state is locally asymptotically stable for less than0( ); that is, in the yellow region. The horizontal dotted line is the limit of0( ) as goes to negative in…nity (the limit is 5 ). For a = 1; the criticalvalue is0
( a ) = 6 ' 18:85:
It is seen in Figure 4(A) that the steady state loses its stability at the redpoint on the stability switching curve (i.e., = 6 ) and in Figure 4(B) that itbifurcates to a limit cycle as a value of becomes larger than the critical value.

(A) Stability region(B) Bifurcation diagram

Figure 4. Stability switching and the birth of a limit cycle.

4

Solow Model with Two Di¤erent Delays

This section is also divided into three subsections. The stability switching curvewith two delays is analytically obtained in Section 4.1. The stability index on
the stability switching curve is determined in Section 4.2. Finally, in Section
4.3, three examples numerically con…rm the analytical results.

4.1

Construction of the Stability Switching Curve

We now consider the general case of 1 > 0; 2 > 0 and
we restate the delay capital accumulation equation:

1 6=

2 . For convinence,_k(t)= sf (k(t

2 ):

(22)

1 ))

(n + )k(tTo study the stability change when 1 and 2 can vary, we will follow themethodology of Gu et al. (2005) with the use of the stability switching curve.

13

To apply this method, we …rst di¤erentiate dynamic equation (22) with respect
to delay variables and then obtain the linearized version of (22) evaluated at
the equilibrium state k ,
_
k(t)
=

k(t

1)

k(t

2)

(23)

where, from the de…nition of the equilibrium share of capital and from linearized
equation (23),sf 0 (k ) =:With an exponential solution, k(t) = e t u; the characteristic equation is
a( ;

1;

+ a2 ( )e2 ) = 1 + a1 ()e

a1 ( ) =and a2 ( ) =

1

2

=0

(24)

where
:

For each= i!, the term a(i!; 1 ; 2 ) in the complex plane is represented
as the sum of three vectors, 1, a1 (i!)e i! 1 and a2 (i!)e i! 2 , with magnitudes 1, ja1 (i!)j and ja2 (i!)j ; respectively. If these vectors create a triangle(i.e., a(i!; 1 ; 2 ) = 0), then the characteristic equation has a solution for somedelays, 1 and 2 . Since the length of each side of a triangle cannot exceed
the sum of the lengths of the remaining two sides, we have the following threeinequalities:
ja1 (i!)j + ja2 (i!)j

1 and

1

ja1 (i!)jja2 (i!)j

1:

(25)

which are called the triangle conditions. Figure 5 illustrates a triangle thatsatis…es the conditions (25).

Figure 5. Triangle conditions
For= i! with ! > 0; these vectors are
a1 (i!) = i!

and a1 (i!) =
14

i ;
!

their absolute values areja1 (i!)j =!

and ja2 (i!)j =!

;and their arguments arearg [a1 (i!)] =3:
2

and arg [a3 (i!)] =2
The triangle conditions give an interval of !;I = [(1) ; (1 +

) ]:

(26)

For any ! 2 I; the characteristic equation has a pair of pure imaginary roots.
In Figure 5, 1 and 2 represent the internal angles of the triangle and aredetermined by the law of cosine as follows:
#
"
2
2
2
1
1 !
1 (!) = cos
2
!
and
1

2 (!) = cos"!2 + 12 !

It is possible to identify solutions ( 1 ;following equations,

22 ) of a(i!;

2

1;

#

:

2 ) = 0 that satisfy thearg a1 (!)e i! 1 + 2m

1 (!) =

arg a2 (!)e i! 2 + 2n

2 (!) =

and:Two cases appear in these equations, since the triangle can be placed above andunder the horizontal axis. Solving them for 1 and 2 yields, for integers m andn,i1h+ (2m 1)(27)
1 (!)
1 (!; m) =
! 2
and
1 3
+ (2n 1)(28)
2 (!) :
2 (!; n) =! 2
Let B and R be the pair of the delays de…ned as
B=f+
1 (!; m);2 (!; n)j ! 2 I; m; n 2 Zg

R=f

1 (!; m);+2 (!; n)j ! 2 I; m; n 2 Zg:

andThen we have the following:

Theorem 4 The stability switching curves are given as
T = B [ R:(29)

This is the set of all 1 and 2 such that a(i!; 1 ; 2 ) = 0 on which the characteristic equation has a purely imaginary root, = i! with ! > 0:
15

4.2

Stability Index

In order to …nd the directions of stability switches, we …rst computea1 (i!)e i! 1 = i!

(cos ! 1

i sin ! 1 )(cos ! 2

i sin ! 2 )anda2 (i!)e i! 2 =i!

with real and imaginary parts
R1 = Re a1 (i!)e i! 1 =I1 = Im a1 (i!)e i! 1 =sin ! 1 ;!

cos ! 1!

andR2 = Re a2 (i!)e i! 2 =I2 = Im a2 (i!)e i! 2 =!
!

sin ! 2 ;cos ! 2 :Then the stability index is de…ned as
Q = R2 I1

R1 I2

2

=

!2

[sin ! 1 cos ! 2

sin ! 2 cos ! 1 ] :By the addition theorem,
2

Q=

!2

sin [! ( 1

2 )]

(30)

which has the same sign as sin [! ( 12)].
We call the direction of the stability switching curve that corresponds toincreasing ! the positive direction. We also call the region on the left-hand(right-hand) side of the stability switching curve the region on the left (right)
when we head in the positive direction of the curve. We are now ready to mention the stability loss and gain on the stability switching curve. If Q 6= 0, thenwe have the following result from Theorem A.2 of Matsumoto and Szidarovszky(2018):
Theorem 5 Let ! 2 I such that i! is a simple pure complex eigenvalue. Ata point ( 1 ; 2 ), we look toward increasing values of !. As ( 1 ; 2 ) moves fromthe region on the right to the left of the corresponding curve of T , a pair ofeigenvalues cross the imaginary axis to the right if Q > 0: If Q < 0, then the
crossing is in the opposite direction.

16

If Q = 0, then Theorem 5 does not indicate the direction of the stability
switches. This is the case when ! ( 1with an integer k. Then
2) = k! 2=k! 1 ; hence at = i!, the characteristic equation becomeseik

e i! 1 + eik e i! 1 = i!

i!
Notice thateik = cos k + i sin k =(e i! 1 = 0:(31)

1 if k is even,
1 if k is odd.

Introduce the notation,(1) < 0 if k is even,(+ 1) > 0 if k is odd,(cos ! 1

i sin ! 1 ) A(k) = 0:

A(k) =
since(< 1. Then from (31),i!

Separating the real and imaginary parts,
A(k) cos ! 1 = 0
andA(k) sin ! 1 =

!:

Hence
cos ! 1 = 0 and sin ! 1 =

!
=
A(k)

(

1 if k is even,
1 if k is odd,

and therefore,!=(

A(k) = (A(k) = (1) < 0 if k is even,1) > 0 if k is odd.

These values correspond to the end points of the interval I:/ So Q = 0 holds at
the initial and endpoints of the di¤erent segments of T , implying that stability
switch cannot occur inside the segments. Hence the direction has to be the sameas at the segments starting or ending at this point.

4.3

Three Examples

Solow (1956) con…rms his analytical results by working out three numericalexamples in which the production function is speci…ed as a Leontief type, a
Cobb-Douglas type or a CES type. In the same way, we examine three examples
to see how the two delays a¤ect the balanced growth.
Example 1: Leontief technology.
17

We …rst take = 5 with which the CES production function (9) might
approximate the Leontief function. The corresponding stability switching curveis described by the blue-red segments in Figure 6(A) in which the blue segments
belong to B of (29) and so do the red segments to R. More precisely,
B = B 0 [ B 1 and R = R1 [ R2
where the four segments are de…ned as follows, as the vertical shift parametern is …xed at n = 0 and the horizontal shift parameter m increases from 0 to 2
with a unit increment,Bi =+1 (!; i);2 (!; 0)for i = 0; 1

(32)

Rj =

1 (!; j);+2 (!; 0)for j = 1; 2:

(33)

and
B 0 is the left-most blue segment and R1 is next to the right of B 0 . The
arrows on the blue and red segments indicate the positive directions of thecorresponding segments.
Let us calculate the stability index on the segments. By (30), the stability
index on the segments B; denoted by QB , is
2

QB =

+
1 (!)

sin !

!2

2 (!)

:

(34)

By (27) with m = 0; 1; 2 and (28) with n = 0;
sin !

+
1 (!)

2 (!)

=

sin [ 1 (!) +where the inequality is due to 0
1 (!) + 2 (!)
index on the segments R, denoted by QR ; is2 (!)]

0: Similarly, the stability

2

QR =

!2

sin !

1 (!)

+
2 (!)

(35)

where
sin !

1 (!)

+
2 (!)

= sin [ 1 (!) +2 (!)]

0:Hence we have QB < 0 on the blue segments and QR > 0 on the red segments
in Figure 6(A) in which the blue Ri and Li for i = 0; 1 denote the region on
the right and the region on the left with respect to the blue segments and so do
the red Rj and Lj for j = 1; 2 with respect to the red segments. The stability
switching curve divides the nonnegative region of ( 1 ; 2 ) into two. When a pairof ( 1 ; 2 ) moves from the lower region to the upper region, the stability is lost,
according to Theorem 5.
Let next check the stability change at the connecting points of B i and Rj .
It is seen in Figure 6(A) that B 0 and R1 start at point S0 . Hence, it is thestarting point of both segments,
S0 = ( a1 ;a2 ) = B 0 \ R1

18

with
a
1 ' 15:87,

a2 ' 15:87 andaa
1 = 2 = 1:

In the same way, it is observed that point E1 is the ending point of segments
B 1 and R1 ;E1 = b1 ; b2 = B 1 \ R1withb
1 ' 46:64,

b2 ' 15:55 andbb
1 = 2 = 3:

and point S1 is the starting point of segments B 1 and R2 ;S1 = ( c1 ;c2 ) = B 1 \ R2

withc
1 ' 79:37,

c
2 ' 15:87 andcc
1 = 2 = 5:

The ratio 1 = 2 is an odd integer so that Q = 0 and the direction of stability is
the same as that on the segment B i or Rj :8 Hence, when a value of 2 increases
along the vertical dotted line at a1 ; b1 or c1 and then crosses the stability
switching curve from below, stability is lost.
We perform a numerical simulation to see what dynamics the delay system(22) can generate. The equilibrium point is locally asymptotically stable for( 1 ; 2 ) below the curve and unstable for the pair above. At point A withAA
1 = 20 and 2 ' 15:9 on the red segment R1 ; stability is lost. The bifurcation
diagram with respect to 2 is illustrated in Figure 6(B) in which 1 is …xed at A1
min
and 2 increases from min
to maxwith an increment of ( max2222 )=500 along
the vertical dotted black line wheremin= 14:9 '
2

A
2

1 and

max
= 20:5 '2A
2 + 4:6:

max
For each value of 2 2 min; the delay dynamic equation (22) runs
2 ; 2
for 0
t
T (= 20; 000) and the local maximum and minimum values of
500 t T are plotted against the selected
2 obtained from the data for T
value. As can be seen, the equilibrium point is locally asymptotically stable forAAA
2 < 2 , loses its stability at 2 = 2 and bifurcates to a limit cycle for 2 > 2 :9B
It is also seen that a cycle-doubling bifurcation occurs at 2 ' 18:44 and so
8 Notice that at S ; the critical values are the same, a = a : Equation (21) can determine011
those values for = 5 and ` = 0:
9 In a discrete dynamical system, Figure 6(A) is called the period-doubling bifurcation inwhich the period of a periodic cycle is doubled. In a continuous dynamical system, a limit
cycle having two extrema is changed to a limit cycle having four extrema, implying the numberof cycles is doubled. Hence we call it "cycle-doubling."

19

does a cycle-halving bifurcation atC
2 ' 20:02 when(A) Stability Region10
2 further increases.

(B) BIfurcation diagram

Figure 6. Leontief technology
Example 2. Cobb-Douglas technology
We take = 1=10 with which the CES production function (9) might approximate the Cobb-Douglas function. It is seen in Figure 7(A) that the stability
switching curve takes a wave-like shape and its amplitude becomes larger than
the tipple-like curve shown in Figure 6(A). Although the big waves are repeated,
we limit value of 1 up to 50 in order to con…ne attention to the …rst wave. As
in Example 1, we denote the blue segments by B i for i = 0; 1 and the red segment by R1 . Applying the formulas in (34) and (35), we …nd that QB < 0 and
QR > 0: The blue Ri and Li above and below the curve denote the regions on
the right and the left along the blue segment while the red R1 and L1 denote
the right and left region along the red segment. According to Theorem 5, thestability is lost when a pair of ( 1 ; 2 ) crosses the blue segments from below
to above. Although the red segment changes its slope from positive-sloping tonegative sloping, the region on the left is located above the positive sloping partand on the right of the negative sloping part. Hence, the stability is lost when
the pair of the delays moves from below to above or from left to right. The
starting point of B 0 and R1 is denoted as S0 witha
1 ' 24:44;

a2 ' 24:44 andaa
1= 2 = 1

and the ending point of B 1 and R1 by E1 withb
1 ' 34:72;

b
1 ' 11:57 andbb
1 = 1 = 3:

Stability is lost when the pair of the delays passes these connection points frombelow to above.
1 0 These bifurcation values are numerically obtained.

20

At point A on the blue segment B 0 , the boundary values are
1 =A
1 = 20 and

2 =A
2 ' 22:668

at which the equilibrium point loses stability. A limit cycle emerge from increasing 1 above A
1 as shown in Figure 7(B) in which the delay dynamic equation(22) runs for 0 t T (= 50; 000) with the initial function '(t) = k
0:1 fort 0 and (k(t); f (k(t))) for T 1000 t T is plotted. We take = A
2 + 0:1
A
for the small cycle, = A
+
0:5
for
the
medium
cycle
and
=
+
1
for the2
2large cycle. The cycle is getting larger as the value of / increases. For a muchlarger value than A
2 + 1, the nonnegative constraint of the per-capita capital
may be violated.

(A) Stability Region(B) Limit cycles

Figure 7. Cobb-Douglas technology
Example 3. CES technology with

= 1=2

The value of is increased to 1=2. This is the case considered by Solow (1956)
in which two possibilities, endogenous steady-state growth and stable balanced
growth, are numerically con…rmed. We study another dynamic behavior of adelay CES economy. As can be seen in Figure 8(A), the larger value of makes
the red segment of the stability switching curve more winding than that in
Figure 7(A). The region division and the stability-loss-and-gain are the same as
those in Example 2. The starting point S0 and the ending point E1 have thefollowing coordinates,S
1 ' 29:72;

S
2 ' 29:72 andSS
1= 2 = 1E
1 ' 32:03;

E
2 ' 10:68 andEE
1 = 2 = 3:

and

21

Stability is lost when a pair of ( 1 ; 2 ) crosses the blue segments from below toabove. In addition, the more winding red segment generates a new phenomenon,
multiple stability switches. The value of 2 is increased along the dotted verticalAline standing at A
1 : The pair of ( 1 ; 2 ) crosses the negative-sloping part of the
red segment at point A from region R1 to region L1 that means the lower left
region. Since the stability index is positive (i.e., QR > 0), Theorem 5 implies
the loss of stability. With further increasing 2 ; the pair passes through point
B on the positive-sloping part from below to above. Along the red segment,
this is a movement from the left region (i.e.,. L1 ) to the right region (i.e., R1 ),implying the gain of stability. Further increasing 2 brings the pair to point
C on the ‡atter positive-sloping part, above which the upper left region L1 is.
Going cross the point from below to above means the movement from the rightregion (i.e., R1 ) to the left region (i.e., L1 ). With the positive stability index,
we have the loss of stability again. No stability switch occurs afterward.
Let us see what dynamics the delay system (22) brings about when the stability is violated. It is numerically checked that an unstable trajectory exhibits
expanding oscillations after point A until point B. Hence k(t) sooner or later
becomes zero, implying that the output production ceases and capital accumulation is terminated. It is also con…ned that the equilibrium point bifurcates toa limit cycle after point C (that is, for 2 > c2 ) as shown in Figure 8(B).

(A) Stability region(B) Bifurcation diagram

Figure 8. CES technology with

5

= 1=2

Concluding Remarks

This study characterizes the relation between the elasticity of factor substitution and the stability/instability of the steady state of a Solow model with two
delays. Our main …nding is that the multiple stability switching can occur if

22the elasticity is high whereas multiple cyclic dynamics could be possible if theelasticity is low. The existence of more complex dynamics could be possible,
however, not con…rmed. It is interesting to apply the delay approach to different one-sector models of economic growth such as the Diamond overlapping
generation model and the Keynesian model as well as to a two-sector growthmodel.

Appendix
In this Appendix, following Hicks (1963), we show the equivalence of thetwo de…nitions of the elasticity of substitution by Hicks (1932) and Robinson(1933). We transform the de…nition of R to the that of H under the conditions:only two inputs, homogeneity of degree one and perfect competition. With the
production function with the degree one homogeneity and perfect competition,
Euler’s theorem says that the factors of production are paid according to their
marginal productivities and then output exactly covers the factor payment,
Y = FK K + FL L

(A-1)

where Y = F (K; L). Partially di¤erentiating (A-1) with respect to K and L
presents
FKK K + FLK L = 0
(A-2)
and
FLL L + FKL K = 0:

(A-3)

From (2), the formulation of Robinson’s de…nition is rewritten,
R =d (log K
d (log FL

dK=K dL=L
log L)=
log FK )dFL =FL dFK =FK(A-4)

The denominator of (A-4) can be developed as follows:
dFL
FL

dFK
FK

=
=

FLL dL + FLK dK
FKK dK + FKL dLFLFKFKK FL FLK FK
FLL FK FKL FL
dL
dK
FK FL
FK FL

FLL from (A-3) and FKK from (A-2) are substituted in the second line of theabove equation to obtain
FKL
FK FL

(Fk K + FL L)

dK
K(Fk K + FL L)

dL
L

=

FKL
Y
FK FL

dK
K

dL
L

:

The last term is replaced with the denominator of (A-4) and then we arrive at
Hicks’de…nition,
FK FL
= H:
R =
FKL Y

23

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