Delay Differential Neoclassical Growth Model∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Abstract
This paper develops a continuous-time neoclassical growth model withtime delay. Despite of its simple structure, the resulting dynamic system shows emergence of erratic fluctuations in the capital accumulation
process when the production function is unimodal and the delay in production is explicitly considered. It complements the seminal paper ofDay (1982) in which a discrete-time neoclassical growth model displayed
chaotic behavior for some configurations of the propensity to save, thegrowth rate of labor and the capital depreciation rate. Our analysis has
at least two implications. First, nonlinearities and delay matter for a birthof aperiodic fluctuations of national product in the continuous-time model.
Second, comparing the effects caused by two different time delays, fixedand continuously distributed time delays, reveals that the continuous-timemodel with the former can generate complex fluctuations while the one
with the latter generates simple fluctuations when the shape parameterof the density function is small and complex fluctuations when large.

Keyword: Neoclassical growth model, continuously distributed timedelay, fixed time delay, complex dynamics, period-doubling bifurcation

JEL: E13, E32, C62∗ The authors highly appreciate financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 21530172), Chuo University (Joint
Research Grant 0981) and the Japan Economic Research Foundation. They are also gratefulto K. Ishiyama and M. Kobayashi for their supporting computational studies. The paper was
prepared when the second author visited the Department of Science and Engineering of Chuo
University. He appreciated its hospitality over his stay. The usual disclaimer applies.
† Department of Economics, Chuo University, 742-1, Higashi-Nakano, Tokyo, 192-0393,
Japan. akiom@tamacc.chuo-u.ac.jp +81-426-74-3351 (tel); +81-426-74-3452 (fax)‡ Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ,
85721-0020, USA. szidar@sie.arizona.edu; +1-520-621-6557 (tel); +1-520-621-6555 (fax).

1

1

Introduction

In his seminal papers, Day (1982, 1983) examines two conventional dynamic
models, one of neoclassical economic growth and the other of productivity andpopulation growth, to show emergence of complex behavior from a quite simple
economic structure when sufficient nonlinearities are present. Both models areconstructed in discrete-time and have the mound-shaped production function
representing the negative effects of pollution caused by increasing capital. It isthen analytically and numerically demonstrated that the models generate not
only periodic cycles but also chaotic behavior for certain values of the shapeparameter of the production function. Each paper has been one of the mostinfluential papers in the study of nonlinear economic dynamics for the last three
decades. Since the publications of these papers, a lot of efforts have been devoted
to deepen the understanding of economic complexity including chaotic behavior,multistability, basins of attractions and so on. Overviews of early contributions
and recent developments in nonlinear economic dynamics are found in, to namea few, Day (1994), Puu (2003) and Bischi et al. (2010).
In discrete-time analysis, there is a clear and simple criteria such as the"period-three condition" of Li and Yorke (1975) to detect chaos, which can beapplied to first-order nonlinear difference equations. In consequence, discretetime chaos has numerous applications to economic models, as shown, for example, in the papers collected in Rosser (2004). On the other hand, the economic
applications of continuous-time chaos are so far limited. This may be due tothe facts that there are no general criteria to establish the presence of chaos incontinuous-time analysis and that a continuous-time system must be at least
three dimensional. As a result, advanced analytical and numerical considerations are required to find chaos in continuous models.
Despite of this inconvenience, this paper aims to reconsider emergence oferratic and unstable fluctuations in a delay continuous-time neoclassical growthmodel with the following four reasons. The first reason is concerned with continuous time scales and is due to Gandolfo (1997) who raises a question on
the choice of continuous or discrete time scales in the construction of dynamic
models and gives eight reasons to be in favor of the use of continuous models
including the following: it is natural to treat economic phenomena as continuous, provided that the variables to be examined in the dynamic models are theoutcomes of a great number of decisions taken by different agents at different
points of time. This is in particular a case in the aggregate model. The second
is concerned with a delay in production process. In the cases of most models
discussed earlier in the literature, it is assumed that each economic agents has
instantaneous information about its own as well as its rivals’ behavior. This
assumption has mathematical convenience but does not fully describe real economic situations in which there are always time delays between the times when
information obtained and the times when the decisions are implemented. The
third is concerned with nonlinearities. If the analysis is confined to a neighborhood of a stationary point, then linear systems are appropriate. However if
certain "global" phenomenon is considered, then nonlinear behavior is responsible for the endogenous dynamics. Finally the fourth reason is concerned withthe use of the neoclassical model and similar to the argument of Day (1983).
Namely, it has the simplest possible framework and still provides a convenient
starting point for an investigation of continuous complex dynamics. In addition2

to these, we have an academic curiosity to find economic circumstances underwhich continuous models give rise to similar dynamic behavior as discrete models. Similarly to the spirit and function form of the discrete-time neoclassicalmodel of Day (1982), continuous time scales, nonlinearity and delay are assumed
in this paper.
The structure of the paper is as follows. Section 2 gives a brief overview of theneoclassical growth model and discusses the fact that the model is globally stable
if its dynamics is constructed in continuous time while the discrete-time model
possesses the possibility of chaotic outcomes when the stationary state is locallyunstable. The continuous-time neoclassical growth model can be transformed
into a delay differential system which is also capable of producing complex
dynamics. Section 3 introduces continuously distributed time delay and Section
4 replaces it with fixed time delay. Section 5 compares the effects caused bythese two different types of delays. Finally some concluding remarks are given
in Section 6.

2

Neoclassical Growth Model

This section serves as a short summary of the neoclassical aggregate modelof growth, which can be traced back to Solow (1956) and Swan (1956). An
enormous amount of work carried out since then. The main interest has beenon two questions: whether full employment growth is possible and whether suchgrowth is stable. Affirmative answers are given under various conditions. The
turning point is Day’s (1982) model showing the occurrence of chaos.
This section is divided into two parts: First the continuous-time version isbriefly discussed to show asymptotic stability of the steady state and then thediscrete-time version is abstracted. Erratic fluctuations are demonstrated when
the steady state is locally unstable and the production function involves strong
nonlinearities.

2.1

Continuous Time Scales

In its simplest version, the neoclassical growth model assumes that the economy
has one sector producing national product Y with a production function,
Y (t) = F (L(t), K(t))(1)

where L(t) is labor, K(t) is capital and t denotes time. F is assumed to beconventional implying that it is constant returns to scale (i.e., F (αL, αK) =αF (L, K) for α > 0) and the two factors are indispensable for production (i.e.,L(t) > 0 and K(t) > 0). Denoting consumption by C(t) and investment by I(t),the equilibrium in the output market is described by the supply-equals-demand
relation
Y (t) = C(t) + I(t).
(2)
We assume that the amount of consumption is a constant fraction of net income,C(t) = (1 − s)(Y (t) − µK(t))(3)

where s ∈ (0, 1) is the average propensity to save and µ ∈ (0, 1] is the depreciation ratio of capital. There are two sources of dynamics: investment and labor3

supply. The amount of gross investment I(t) is equal to the sum of the change
in capital stock and the amount of capital depreciation at each instant of time,
I(t) = K̇(t) + µK(t),(4)

where the dot over variables denotes time derivatives, e.g., K̇(t) = dK(t)/dt.
We further assume that labor grows at a (exogenously determined) positive
constant rate n > 0,
(5)
L(t) = L0 ent
where L0 is the amount of labor available at t = 0.
Substituting equations (1) and (4) into (2) gives
F (L(t), K(t)) = C(t) + K̇(t) + µK(t).
Dividing both sides by L(t) > 0 and introducing the per-capita variables, c(t) =C(t)/L(t) and k(t) = K(t)/L(t), we obtainf (k(t)) = c(t) +K̇(t)+ µk(t)L(t)(6)

where the assumption of constant returns to scale implies that we can write theproduction function in the "intensive" or "per-capita" formKf (k) = F 1,.
L
Dividing both sides of equation (3) by L(t) givesc(t) = (1 − s)(f (k(t)) − µk(t)).
Differentiating k(t) with respect to t and noticing that the growth rate of labor
is n, we obtain, after arranging terms,
K̇(t)= k̇(t) + nk(t).
L(t)
Combining the last two equations with (6), we have the fundamental dynamic
equation of the neoclassical aggregate growth model:k̇(t) = sf (k(t)) − δk(t) where δ = n + sµ.

(7)

We now impose the conventional assumptions (i.e., Inada conditions) on the
per-capita production function f(k):Assumption 1.
(1) f ′ (k) > 0 and f ′′ (k) < 0 for all k ≥ 0;(2) f (0) = 0;
(3) lim f ′ (k) = ∞ and lim f ′ (k) = 0.
k→0

k→∞

4

Under these conditions, the continuous-time neoclassical growth model (7)has a steady state k∗ > 0 which is the unique positive solution of equationsf(k∗ ) = δk∗ .
Outside this steady state we have two possibilities:k̇ > 0 for k0 < k∗ or k̇ < 0 for k0 > k∗(8)with k0 = k∗ and k0 = 0. The directions of inequalities of k̇ in (8) imply the
local asymptotical stability of the positive steady state. This classical result iswell known from the literature.

2.2

Discrete Time Scales

In this section, we transform the continuous model to the discrete time model
where t now refers to period t. The behavioral specifications (1), (2) and (3)are not affected by changing time scales. On the other hand, dynamic specifications (4) and (5) should be modified accordingly. The labor supply equationis changed to(9)L(t) = (1 + n)t L0
where the notation n is used to denote the growth rate of labor. A law of motion
for the capital stock in discrete time is obtained by replacing K̇(t) in equation(4) by K(t + 1) − K(t), so
K(t + 1) − K(t) = I(t) − µK(t).

(10)

Substituting (3) and (10) into the equilibrium relation (2) and re-arranging
terms yieldsY (t) = K(t + 1) − (1 − sµ)K(t).
Dividing both sides by L(t) and using relations y(t) = f(k(t)) and L(t +1)/L(t) = 1+n yield the discrete-time fundamental dynamic equation of capital:k(t + 1) =1
(sf(k(t)) + (1 − sµ)k(t)) ≡ g(k(t)).
1+n

(11)

Given the initial condition k0 , this first-order difference equation determines theentire time path of the capital stock. The conventional neoclassical assumptions(i.e., Assumption 1) ensures that there is a positive and stable steady stateks > 0 such that ks = g(ks ) and all trajectories monotonically converge to ks
as t → ∞ for all k0 > 0. As a benchmark, given the well known Cobb-Douglas
production function F (L, K) = AK a L1−a with A > 0 being the technology
parameter and a ∈ (0, 1), the elasticity parameter, we have its intensive formf (k) = Aka . In this case it can be confirmed that the steady state is ks =[sA/(n + sµ)]1/1−a which is locally asymptotically stable.
Day (1982) reconstructs this neoclassical theory of capital accumulation under circumstances in which productivity of the production function is reduced
by a pollution effect caused by increasing concentration of capital. Applying
the nonlinear dynamic theory, he demonstrates that erratic behavior exhibiting
wandering and sawtooth patterns can emerge when sufficient nonlinearities are5present. In particular, in contrast to the conventional neoclassical assumptions,the per-capita production function is modified to have the formf (k) = Aka (1 − k)b(12)with A > 0 and f ′ (k) < 0 for large k. The factor (1 − k)b reflects the influence ofpollution on per-capita output. When the capital intensity increases, pollution
increases as well. However for a small enough value of b, the pollution effectis so small that the modified production function behaves like the conventional
production function. When b is relatively large, the pollution effect reduces
national output. Substituting the new production function into (11), we havethe modified fundamental equation of capital accumulation:k(t + 1) =1 − sµsAk(t)a (1 − k(t))b +
k(t).
1+n
1+n

Consider a simplification by assuming that a = b = 1. Introducing the newvariable z = Bk withsA
B=
1 + (A − µ)s
reduces the fundamental equation to the logistic map,z(t + 1) = θz(t)(1 − z(t))
where
θ=

(13)

1 + (A − µ)s.
1+n

A positive fixed point

θ−1,θ
is obtained for θ > 1. The dynamics of k(t) generated by (11) with the modified production function is equivalent to the dynamics of z(t) generated by thelogistic map (13). The dynamic structure of (13) has been extensively studied
and it is now well-know that equation (13) can generate a wide variety of dynamics ranging from monotonic convergence to chaotic behavior, depending on
the value of θ. The factor θ usually varies between zero and four. In summary,the discrete-time neoclassical growth model (13) can generate the following dynamics:
zs =

(1)
(2)
(3)
(4)

3

monotonic convergence to zero for 0 < θ ≤ 1,
monotonic growth and convergence to zs for 1 < θ ≤ 2,
oscillatory convergence to zs for 2 < θ ≤ 3,
continued oscillations around zs for 3 < θ ≤ 4.

Growth Model with Continuously Distributed
Time Delay

It is demonstrated in the first half of the last section that the continuous-time
neoclassical growth model is definitely stable. It is also demonstrated in the latter half that the discrete-time neoclassical model may induce erratic dynamics
in the capital-labor ratio under strong nonlinearity presented by the larger value
6

of θ. There is a sharp difference between dynamics generated by the continuousand discrete models. To discuss emergence of irregular fluctuations in the continuous model, it may be helpful to take account two facts; one is that a simple
nonlinear delay differential equation such as the Mackey-Glass equation (1977)
may display complex dynamics involving chaos and the other is that there isalways a time delay due to the decision making process and the decision implementation. In this and next sections, we will confine our attention to delays in
the capital accumulation process. Using the mound-shaped production functionas, for example, in the case of the modified production function (12), we take aclose look at destabilizing effects caused by delays in production.
If a production delay is present and the delay is known or certain, then the
production function is written asy(t) = f(k(t − τ ))where τ ≥ 0 is the delay. We will examine the case of known delays in the next
section and inquire into the case of unknown delay in this section. A convenient way to introduce unknown or uncertain delay into the model is offeredby considering it as a random variable and replacing it by its expected value.
If the expected per-capita capital stock at time t is denoted by ke (t) and isbased on the entire history of the actual capital stock from zero to t, then thecontinuous-time dynamic system (7) with delay in production can be rewritten
as a Volterra-type integro-differential equation:k̇(t) = −αk(t) + βke (t)(1 − ke (t)),
ke (t) =

 t
0

(14)
ω(t − σ, T, m)k(σ)dσ,

where α = δ, β = sA and the weighting function is defined as 1 t−σ− Tif m = 0, Te
ω(t − σ, T, m) =
 1 m m+1 (t − σ)m e− m(t−σ)
Tif m ≥ 1
m! T

(15)

with τ = t − σ. Here the shape parameter m is a nonnegative integer and T isa positive constant, which is associated with the length of the delay. If m = 0,then weights are exponentially decreasing with the largest weight given to themost current data. If m ≥ 1, then the weighting function has a bell-shaped
curve implying that the most current data has zero weight, the weight increases
to a maximal value at t − σ = T and decreases thereafter. A steady state of(14) is obtained by setting k̇(t) = 0 and k(t) = ke (t) = k∗ ,
k∗ =

β−α
< 1.
β

To ensure positivity of the steady state, we make the following assumption:
Assumption 2. β − α > 0.

7

To examine local dynamics of the above system in the neighborhood of thesteady state, we consider its linearized version. The approximated nonlinear
term evaluated at the steady state isβd ek (t)(1 − ke (t)) = β(1 − 2k∗ )kδ (t)dke= (2α − β)kδ (t),where kδ = k − k∗ denotes deviation from the steady state. The linearization
yieldstk̇δ (t) = −αkδ (t) + (2α − β)w(t − σ, T, m)kδ (σ)dσ.

0

(16)

At the steady state, we look for the solution in the usual exponential formkδ (t) = eλt uwhich is substituted into the linearized equation (16)λtλtλe u = −αe u + (2α − β)Notice that τ = t − σ, then tw(t − σ, T, m)eλσ dσ

=

0

 t
0

 0

w(t − σ, T, m)eλσ udσ.

w(τ , T, m)eλ(t−τ ) (−dτ )

t

λt

=e

 t

w(τ , T, m)e−λτ dτ0

By letting t → ∞, the integral part converges to
 ∞

−λτ

w(τ , T, m)e

0

with
q=


−(m+1)λTdτ = 1 +
q


 1


if m = 0,m if m ≥ 1.

Substituting this expression into the integral part of the linearized equation,
dividing both sides of the resultant expression by eλt u and arranging terms, we
obtain the characteristic equation

m+1
λT
−(2α − β) + (α + λ) 1 +
= 0.
q

(17)

It has been already shown that the continuous-time neoclassical model without
time delay is locally stable. By continuity, it can be supposed that the delay
model is locally stable if the delay is small enough. We start our examination
with a general result which generates local asymptotical stability regardless ofthe size of the delay and the shape of the weighting function.
8

Theorem 1 The steady state k∗ of the distributed delay differential equation(14) is locally asymptotically stable if |2α − β| ≤ α, regardless of the values ofm and T .
Proof. Notice first that λ = 0 is not an eigenvalue under Assumption 2. Assume
that λ = 0 and Re λ ≥ 0. Then|λ + α| > α and 1 +implying that

λT
>1
q

m+1

λT
> α.
(α + λ) 1 +q

This relation and condition |2α − β| ≤ α imply that the two terms in the left
hand side of the characteristic equation (17) have different absolute values, thus
λ = 0 with Re λ ≥ 0 cannot be an eigenvalue.
Theorem 1 implies that the time delay is harmless to stability if β ≤ 3α. In
order to examine a possibility of stability loss, we make the following assumption, which makes Assumption 2 redundant:
Assumption 3. β > 3α.
The characteristic equation (17) is a polynomial equation with degree n =m + 2 having the forma0 λn + a1 λn−1 + ... + an−1 λ + an = 0
with all real coefficients. Suppose that a0 is positive. Then the Routh-Hurwitz
theorem states that necessary and sufficient conditions for all roots of this polynomial equation to have negative real parts are given by(1) all coefficients are positive, ak > 0 for k = 0, 1, 2, ..., n,(2) ∆1 > 0, ∆2 > 0, ..., ∆n > 0,
where ∆1 , ∆2 , ..., ∆n are the leading principal minors of the Routh-Hurwitz
matrix
a1 a3 a5 a7 ... 0 a0 a2 a4 a6 ... 0  0 a1 a3 a5 ... 0  0 a0 a2 a4 ... 0  .
 0 0 a1 a3 ... 0  ... ... ... ... ... ... 
0 0 0 0 ... an

Since general stability results cannot be obtained analytically, we will specify
the values of parameters and consider special cases.
Case 1 T = 0.
9

T = 0 corresponds to the model without delay. Taking T = 0 in the characteristic equation gives the unique characteristic root,
λ = α−βwhich is negative under Assumption 3. Local stability under the general form of
the production function satisfying Assumption 1 is already shown in (8). Here
we have local asymptotic stability under the specific production function (12)without time delay. This result implies that a strong nonlinearity alone cannot
be a source of complex dynamics in the continuous-time model. This shows asharp contrast to dynamic results in nonlinear discrete-time models, in which astrong nonlinearity itself is a source of complex dynamics.
Case 2 T > 0 and m = 0.
By substituting m = 0 and q = 1 into (17), we obtain the quadratic characteristic equation(18)
T λ2 + (1 + αT )λ + β − α = 0.
Since a0 = T > 0, a1 = 1 + αT > 0 and a2 = β − α > 0, then ∆1 > 0 and∆2 = a1 a2 > 0. Due to the Routh-Hurwitz criterion, we have local asymptotic
stability when the weighting function is exponentially declining.
Case 3 T > 0 and m = 1.
In this case, the characteristic equation becomes cubic in λ,T 2 λ3 + T (2 + αT )λ + (1 + 2αT )λ + β − α = 0.

(19)

All coefficients are positive, a0 = T 2 > 0, a1 = T (2+αT ) > 0, a2 = (1+2αT ) >0 and a3 = β − α > 0. Then the stability condition is that the leading principle
minors are positive,∆1 = a1 > 0,
∆2 = a1 a2 − a0 a3 > 0,
∆3 = a3 (a1 a2 − a0 a3 ) > 0.

The first and third inequalities are satisfied, given the second. We, therefore,
have asymptotical stability if and only if ∆2 > 0, that is,a1 a2 − a0 a3= T (2 + αT )(1 + 2αT ) − T 2 (β − α)
= T 2α2 T 2 + (6α − β)T + 2 > 0.

Let us denote the expression in the parentheses by ϕ(T ). It is quadratic in T
and its discriminant is
D = (β − 2α)(β − 10α).
Under Assumption 3, the first factor is positive, so we can identify the following
three cases.
(i) If 3α < β < 10α, then D < 0 indicating that ϕ(T ) has no real roots.
Consequently ∆2 = T ϕ(T ) > 0 holds for all T > 0, implying asymptotic stability of the steady state.
10(ii) If β = 10α, then D = 0 so ϕ(T ) has only one root (with multiplicity
two)
6α − β1T∗ = −= > 0.
4α2αThus ∆2 > 0 holds for all T = T ∗ , implying asymptotic stability for all T except
a configuration of (α, β, T ) such that αT = 1 and β = 10α.
(iii) If β > 10α, then D > 0 indicating that ϕ(T ) has two real roots, T1∗ and∗T2 such as 0 < T1∗ < T2∗ where
β − 6α ± (β − 2α)(β − 10α)∗T1,2 =.
4α2
It is clear that ∆2 > 0 holds for T < T1∗ or T > T2∗ and ∆2 < 0 for T1∗ <T < T2∗ . Therefore we have asymptotic stability if T < T1∗ or T > T2∗ and local
instability if T1∗ < T < T2∗ . So stability is lost at T = T1∗ and is regained at
T = T2∗ . Notice that (αT1∗ )(αT2∗ ) = 1.
We visualize the analytical results obtained above in Figure 1. Given a valueof α, the whole region of (β, T ) is divided into two parts by the partition curve,∆2 = 0; the gray region is the instability region with ∆2 < 0 and the white
region is the stability region with ∆2 > 0. The red point on the ∆2 = 0 curve
implies that the characteristic roots are multiple for β = 10α and T = 1/α. From
the above analysis, we can draw the following conclusion in the case of m = 1. If
β < 10α, then the stationary point is locally asymptotically stable regardless ofthe value of T. Returning to the definitions, the value of α = n + sµ depends onmodel parameter values and β = sA is the product of the marginal propensity
to save and the technology parameter. Since we need a large value of thetechnology parameter to reverse this inequality, the high technological progress
may be a source of instability. Assume next that β > 10α. The delay has to be
sufficiently small or sufficiently large to guarantee local asymptotical stability.
For example, when β = 12α, the threshold values of T are
√
√
3− 5
3+ 5
1
1
< and T2∗ =
> ,
T1∗ =
2α
α
2α
α
both of which are indicated by the two black points on the separation curve∆2 = 0 in Figure 1. The stationary point is locally unstable for T in the shaded
region (i.e., T1∗ < T < T2∗ ) and stable for T in the white region (i.e., T < T1∗ or

11

T > T2∗ ).

Figure 1. Stability and instability regions when m = 1
We now return to the characteristic equation (19) and show the possibility ofthe birth of a limit cycle at the threshold values T = T1∗ and T = T2∗ by applying
the Hopf bifurcation theorem. According to the theorem, we can establish theexistence of a cyclic solution if the cubic characteristic equation has a pair ofpure imaginary roots and the real parts of these roots change signs with abifurcation parameter. At the critical values where ∆2 = 0 or a3 = a1 a2 /a0 ,the characteristic equation can be rewritten asa0 λ3 + a1 λ2 + a2 λ +which factors as

a1 a2
= 0,
a0




a1 
λ+
a0 λ2 + a2 = 0
a0

showing that one of the characteristic roots is real and negative,
λ1 = −

a1
2 + αT=−<0a0
T

and the other two are pure imaginary,
λ2,3 = ±i



a2
= ±i
a0



1 + 2αT= ±iγ.
T2(20)

In order to apply the Hopf bifurcation theorem, we need to check whether
the real part of the conjugate complex root changes its sign as the bifurcation
parameter passes through its critical value. Selecting T as the bifurcation parameter, assuming λ = λ(T ) and then differentiating the characteristic equation
with respect to T, we obtain, after arranging terms, the derivative of λ(T ),
2 T λ3 + (1 + αT )λ2 + αλdλ.
=− 2 2dT
3T λ + 2(2T + αT 2 )λ + (1 + 2αT )12Using the facts that the terms with λ are pure imaginary and the constant terms
are real gives






2 T λ3 + (1 + αT )λ2 + αλd(Re λ)= Re − 2 2dT
3T λ + 2(2T + αT 2 )λ + (1 + 2αT )
λ=±iγ
λ=±iγ

=

2(1 + Ti∗2 γ 2 )(1 − αTi∗ ),∗(1 + 2αTi − 3Ti∗2 γ 2 )2 + (4Ti∗ + 2αTi∗2 )2 γ 2

where both the numerator and the denominator of the first factor are positive.
Therefore we haved(Re λ)≷ 0 if 1 ≷ αTi∗ for i = 1, 2.
dT
λ=±iγThese inequalities imply that the roots cross the imaginary axis at iγ from left
to right as T increases from T1∗ < 1/α (i.e., stability is lost) and from rightto left as T increases from T2∗ > 1/α (stability is regained). In Figure 1, thestability loss occurs when T enters the gray unstable region from below and thestability regain occurs when T re-enters the white stable region from below.
In order to find what kind of dynamics is generated for T ∈ (T1∗ , T2∗ ), we perform numerical simulations. To this end, we transform the integro-differential
dynamic system to the three-dimensional dynamic system of ordinary differential equations:k̇(t) = −αk(t) + βke (t) [1 − ke (t)] ,k̇e (t) =1(x(t) − ke (t)) ,T1(k(t) − x(t)) .
T
The first equation is the same as the first equation of (14). With m = 1, theweighting function isẋ(t) =

 t  2t−σ1(t − σ)e− T dσ,k (t) =
T0eand the second equation is obtained by differentiating both sides of the above
equation with respect to t and introducing the new variable t
1 − t−e ee T k (σ)dσ.
x(t) =T
0
The third equation is obtained by differentiating x(t) with respect to t. Needlessto say, the characteristic equation of this 3D system is identical with (19).
Returning to Figure 1, taking β = 12α with α = 1 and increasing T froma little bit smaller value than T1∗ to a little bit larger value than T2∗ yield thebifurcation-like diagram as is presented in Figure 2(A). It is shown that thestationary point k∗ is asymptotically stable for T < T1∗ and T > T2∗ and it
bifurcates to cyclic behavior if T is between T1∗ and T2∗ . For each value of T,
the dynamic 3D system is solved and we measure a time series of k(t), ke (t)13and x(t) for 3000 ≤ t ≤ 4000. The extreme values (i.e., local maximums andlocal minimums) of the time series k(t) are plotted in blue over each T . The
two (upper and lower) blue curves mean that a periodic cycle generated foreach T ∈ (T1∗ , T2∗ ) has one maximum and one minimum. In Figure 2(B) we take
Ta = (T1∗ +T2∗ )/2 and depict the periodic cycle in blue in the phase plane of ke (t)and k(t), which describes the dynamic flow based on the measured data. It canbe seen that points kH and kL are the maximum and minimum values of the cycle
when T = Ta . Two observations can be made: first, there are two points for eachT ∈ (T1∗ , T2∗ ) implying that the trajectories converge to a periodic cycle having
one maximum and one minimum; second, some segments of the trajectories
enter the negative region of k. One way to avoid such economically unfavorable
phenomena that the capital dynamics might lead to negative quantities is tomodify the capital dynamic equation in such a way thatk̇(t) = −αk(t) + βM ax {0, ke (t) [1 − ke (t)]} .
The simulation results with this modified dynamic equation are obtained in
the same way and are depicted as the red curves in Figure 2(A) in which the
trajectories of capital stay in the positive region for T ∈ (T1∗ , T2∗ ). The red
cycle in Figure 2(B) is in the first quadrant of the phase plane and has the
maximum kh and the minimum kl . The point kj for j = H, h, ℓ, L in Figure2(A) corresponds to the same point in Figure 2(B). We will return to the second
observation in the final section.

(A) Bifurcation diagram(B) Periodic cycles/ for T = Ta

Figure 2. Birth of cyclic behavior when stability is lost
Case 4 T > 0 and m = 2.
Substituting m = 2 into the characteristic equation (17) yields the following
fourth-order equation
a0 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0
where
a0 = T 3 > 0, a1 = T 2 (6 + αT ) > 0, a2 = 6T (2 + αT ) > 0,
a3 = 4(2 + 3αT ) > 0 and a4 = 8(β − α) > 0.
14

All coefficients are positive. In the case of a fourth-order equation, the basic
Routh-Hurwitz matrix is
a1 a3 0 0
 a0 a2 a4 0  0 a1 a3 0 
0 a0 a2 a4
with leading minors∆1 = a1 > 0,
∆2 = 2T 3 (3T 2 α2 + 18T α + 32) > 0,
∆3 = 8T 3 64 + 12T (14α − 3β) + 12T 2 α(6α − β) + T 3 α2 (10α − β)  0,

∆4 = ∆3 a4 > 0 provided that ∆3 > 0.

Therefore ∆3 > 0 suffices to achieve local asymptotic stability. Notice that
∆3 = 0 if the cubic equation in T in the braces is equal to zero. The discriminantof this cubic equation isD= −4A31 A3 + A21 A22 − 4A0 A32 + 18A0 A1 A2 A3 − 27A20 A3= 442368α3 (β − 2α)2 (β − 6α)

where A0 = 64, A1 = 12(14α − 3β), A2 = 12α(6α − β) and A3 = α2 (10α −β). The cubic equation ∆3 = 0 can be reduced to 4(−2 + αT )2 (4 + αT ) = 0
if β = 6α. It is then clear that the cubic equation has the unique root withmultiplicity two for β = 6α and T = 2/α. By taking the same procedure as in
the case of m = 1, we can show that the dynamic system with m = 2 generates
cyclic behavior when the local stability is violated, as illustrated in Figure 3.
It is possible to examine the case with m ≥ 3. However, calculations becomes
longer and more clumsy.

Figure 3. Birth of a limit cycle when m = 2, α = 1, β = 7 and T = 1
15

4

Growth Model with Fixed Time Delay

A fixed time delay is an appropriate approach to deal with the case in which thedelay is certain. If τ represents the time delay inherent in the production process,then the per-capita production function reflecting the influence of pollutions is
described byf(k(t − τ )) = Ak(t − τ )((1 − k(t − τ )).

By replacing ke (1−ke ) in the first equation of (14) by this expression, we obtaina fixed delay differential equationk̇(t) = −αk(t) + βk(t − τ )(1 − k(t − τ )).

(21)

The dynamics of this equation is examined henceforth. A stationary pointof the capital accumulation dynamics is obtained by solving k̇(t) = 0 withk(t) = k(t − τ ) = k. A simple calculation shows that it is identical with k∗obtained under the continuously distributed time delay.
Following the tradition of the study on fixed delay equations, we start fromthe local stability analysis of the stationary state and then consider the question of stability switching of the fixed delay equation (21). The stability of thestationary state depends on the location of the roots of the associated characteristic equation. The stability analysis concerns with whether all roots lie inthe left half of the complex plane. The location of the roots, in turn, depends onthe value of the delay. The stability switching analysis concerns with whetherroots cross the imaginary axis when the delay changes.
For the first purpose, we make a linear approximation of the nonlinear delay
equation (21) to obtain its variational equation,k̇δ (t) = −αkδ (t) + β(1 − 2k∗ )kδ (t − τ )where kδ (t) = k(t) − k∗ is the deviation from the stationary state. This is a
first-order delay differential equation of retarded type. We look for the solution
in the usual exponential form kδ (t) = eλt u. Substituting this expression into
the variational equation and noticing that β(1 − 2k∗ ) = 2α − β, we have thecorresponding characteristic equation,(λ + α) − (2α − β)e−λτ = 0.

(22)

Without time delay (i.e., τ = 0), the characteristic root is λ = −(β − α) < 0.
Thus the steady state k∗ is a stable fixed point which is attained for any nonzero initial point. We can also obtain the following stability result for the fixed
delay model with τ > 0, which corresponds to Theorem 1 obtained in the caseof distributed-delay models:Theorem 2 The steady state k∗ of the fixed delay dynamic equation (21) islocally asymptotically stable if |2α − β| ≤ α, regardless of the value of the fixed
time delay τ .
Proof. Assume to the contrary that for an eigenvalue, Re λ ≥ 0. Thene−λτ= e−(Re λ)τ e−i(Im λ)τ
= e−(Re λ)τ (cos((Im λ)τ ) − i sin((Im λ)τ )) .
16

So e−λτ = e−(Re λ)τ ≤ 1. Notice that since λ = 0,|λ + α| > αand(2α − β)e−λτ ≤ |2α − β| .

Under condition |2α − β| ≤ α, the absolute values of the two terms of thecharacteristic equation are different and thus λ = 0 with Re λ ≥ 0 cannot be an
eigenvalue. This completes the proof.

Theorem 2 ensures that the fixed time delay is harmless if β ≤ 3α. Even
if this sufficient condition is violated, it can be supposed, due to continuity,that k∗ can be still stable for a smaller value of τ . It is also expected that as
the length of the delay increases, stability of the stationary state may change(i.e., stability switch). In order to understand this phenomena, it is crucial todetermine a threshold value of τ for which the characteristic equation has a pairof conjugate pure imaginary roots. Since λ = 0 is not a root of the characteristic
equation, we can assume without loss of generality that λ = iω with ω > 0 is aroot. Substituting it into the characteristic equation, we can write the real andimaginary parts as:
α − (2α − β) cos ωτ = 0,(23)
ω + (2α − β) sin ωτ = 0.
The sum of the squares of the two equations yields
ω 2 = (β − α)(β − 3α)implying thatω+ =

(β − α)(β − 3α),which is real and positive under Assumption 3. Solving the first equation of(23) for τ and substituting ω + into the resultant expression yields the thresholdvalue of time delay for which stability switching may occur


α
cos−1
2α − βτ∗ = ≡ φ(β, α).
(24)(β − α)(β − 3α)In order to observe stability switching, we need to determine the sign of thederivative of the real part of the purely imaginary root. We can select τ as thebifurcation parameter and consider the characteristic equation as a continuousfunction in terms of the delay τ . Differentiating the characteristic equation withrespect to τ yields dλ
= −(2α − β)λe−λτ .
1 + (2α − β)τ e−λτdτFor convenience, we study (dλ/dτ )−1 instead of dλ/dτ . Then we have −1dλ
1 + (2α − β)e−λτ τ=dτ−λ(2α − β)e−λτ
17and from the characteristic equation we obtaine−λτ =λ+α.
2α − β

Thus


Re



dλ
dτ

−1 

=

λ=iω

=
=
=




eλτ
τ
Re
−
−λ(2α − β) λ λ=iω
1 + τ (λ + α)Re −λ(λ + a)λ=iω 2ω (1 + τ α) − τ ω 2 αRe
ω 4 + ω2 α2
1> 0.
ω 2 + α2The last inequality implies that the crossing of the imaginary axis is from theleft to the right as τ increases and thus results in the loss of stability. In otherwords, the fixed delay has a destabilizing effect since the real part turns to bepositive from negative as the delay becomes longer. In summary we have thefollowing:
Theorem 3 Under Assumption 3, the neoclassical growth model with fixed timedelay is locally asymptotically stable when τ < τ ∗ and unstable when τ > τ ∗where the critical level of time delay τ ∗ is defined by


α
cos−1
2α − βτ∗ = 
.
(β − α)(β − 3α)Locally stable and unstable combinations of β and τ are graphed in Figure
4 where α is taken to be 1. The dotted vertical locus, β = 3α, divides the whole(β, τ) region into two parts; the steady state k∗ is definitely locally stable in theleft of the locus due to Theorem 2. Further, the black-bold locus of τ ∗ = φ(β, α)is asymptotic to the β = 3α locus and the horizontal axis and divides the remaining region into two parts: the steady state is locally stable for (β, τ) in
the region below this locus and unstable above. Since this partition curve between the stable and unstable regions is downward-sloping, larger (respectively
smaller) values of the two variables contribute instability (respectively stability)
while a trade-off between the two is necessary to preserve stability. We now turnattention to the instability region above the partition curve. It is colored andshows a bifurcation diagram with respect to β and τ . Different colors indicate
different periods of cycles up to 16. Periodic cycle with a period larger than
16 and aperiodic cycle are colored in red. A period of one cycle is defined bya number of crossing of a time-trajectory with the k = k∗ locus from above
within one cycle (from one peak to the next peak or from one trough to thenext trough). This diagram shows that the delay dynamic equation (21) generates a wide variety of dynamics ranging from a simple limit cycle to chaotic
fluctuations when stability is lost. As is seen shortly, the dynamic equation
gives rise a period-one cycle for a combination of β and τ in the purple region
18

just above the separation curve, a period-two cycle for a combination in thedark purple region above the purple region, a period-four cycle in the thin blue
region and chaos in the red region. The equilibrium point goes to chaos via theperiod-doubling cascade that displays a sequence of bifurcations in which theperiod of the cycle doubles as a parameter combination is changed slightly.

Figure 4. Two-parameter bifurcation diagram of the fixed delay equation (21)To check what dynamics emerges in each different colored region, we take
α = 1, β = 4α, pick up three dotted points, (4α, 2), (4α, 3) and (4α, 4) along thevertical real line in Figure 4 and then perform simulations under these specifications of the bifurcation parameters. These show a sequence of a time trajectoryand (k(t), k(t − τ )) plots of these time delay embeddings for three different values of τ . Notice that k(t − τ ) is the ordinate in both diagrams in each part of
Figure 5 to make comparison easier. When τ = 2, as shown in Figure 5(A), atime trajectory regularly repeats ups and downs and intersects the k = k∗ locus
once from above within one cycle. That is, the plot of k(t) and k(t − τ ) forms a
simple limit cycle which we call a period-one cycle. The dynamic system withany parametric combination in the purple region exhibits the same dynamics.
When τ is increased to 3, then Figure 5(B) shows that regularity of the time
trajectory is distorted and there are three local maxima and three local minima
within one cycle. However it is also observed that the time-trajectory intersects
the k = k∗ locus twice from above implying that a period-two cycle was born
as depicted in the (k(t), k(t − τ )) plane. When τ = 4, as shown in Figure 5(C),the time trajectory exhibits erratic fluctuations implying a very long period orchaos. Accordingly the return map also displays chaotic motions. Although it
is not presented here, we have numerically confirmed that the dynamics withparameters in the red region have initial point dependency so that arbitrarily

19close two initial conditions diverge as time proceeds.

Figure 5(A). Periodic cycle with τ = 2

Figure 5(B). Periodic cycle with τ = 3

Figure 5(C) Aperiodic cycle with τ = 420

To extend the above analysis, we continuously increase the value of τ from
1 to 5 along the vertical real line in Figure 4. When the value of τ passes
the threshold value τ ∗ = φ(β, α), which is approximately 1.29 for the specified
values of α and β, the fixed point loses its stability and bifurcates to a periodic
solution. Simulation results are summarized in a bifurcation-like diagram of k(t)against τ in Figure 6. It is constructed in the following way. For each value of
τ > τ ∗ , the delay equation (21) is numerically solved for 0 < t ≤ 400, given an
initial function ϕ(t) = 0.5 for t ∈ [−τ , 0]. The local maximums and minimumsof the time trajectory for t ∈ [300, 400] are plotted above the selected valueof τ . This diagram exhibits a number of local maximum and local minimum
within one cycle so it is more likely to the Poincare section. As is already seen
above, a time-trajectory has one local maximum and one local minimum forτ = 2 so that there are two points over τ = 2. Two points bifurcates to four
extrema that turn to six extrema, shortly after two more points come in. We
have already checked that the period-two cycle has six extrema when τ = 3 asshown in Figure 6 where the vertical dotted line at τ = 3 crosses six time thered colored graphs. Although the bifurcation of the extremes does not follow
exactly period-doubling, this process repeats itself to generate periodic cycles
with higher periods and then chaotic fluctuations.

Figure 6. Bifurcation diagram with respect to τ , given α = 1 and β = 4α5Comparison of Continuously Distributed andFixed Delays

No matter which time delay is introduced into the neoclassical growth model,
it is found that the delay has a destabilizing effect if Assumption 3 is violated.
In this section, we compare the destabilizing effect caused by the continuously
distributed time delay with the one by the fixed time delay. We have obtained
the instability and stability regions separated by equation (24) under the fixed
time delay. We have also obtained the stability regions divided by ∆2 = 0 whenm = 1 and ∆3 = 0 when m = 2 under continuously distributed time delay.
Repeating the same procedure for m = 3 through 5, we get three more partition
curves that divide the (β, T ) space into the stable and unstable regions. The
21

specified forms of these separation curves, however, are not given here becauseof their too-long expressions. Then given α = 1, the τ ∗ = φ(β, α) curve coloredin red and the five black partition curves with m from 1 to 5 are illustrated in
Figure 7. The four red points there correspond to the combination of (β, T )for which the partition equations have multiple roots, although it is not easy
to recognize that all these separation curves have fork-shaped profiles. It canbe seen that all of the partition curves with continuously distributed time delay
are located in the gray region where the steady state is locally unstable under
the fixed time delay and approaching the partition curve with fixed time delay
from the right as the value of m increases.

Figure 7. Partition curves of continuously distributed and fixed time delays
It is natural to guess that the distributed stable region becomes graduallysmaller and converges to the fixed stable region when m tends to infinity. Thisis confirmed to be correct if we notice the properties of the weighting function.
For m ≥ 1, it takes a bell-shaped profile. As m increases, the function becomes
more peaked around t − σ and tends to the Dirac delta function. For sufficientlarge m, the weighting function may be regarded as very close to the Diracdelta function and the dynamic behavior under the continuously distributed
time delay is very similar to the one under the fixed time delay. We can explain
this phenomenon mathematically. Given m ≥ 1, the characteristic equation(17) under continuously distributed time delay can be written as
2α − βm 
 = 0.
λ+α− 
T
T
1+ λ
1+ λ
m
m
As m → ∞, it converges to
λ + α − (2α − β)e−T λ = 0
which is the characteristic equation (22) of the fixed time delay if T is replaced with τ . In short, under continuously distributed time delay, although
we comprehensively use the delayed or past data of capital, the stability regionis sensitive to the shape of the weighting function. Increasing the value of m22

strengthens the destabilizing effect caused by the distributed delay in a sensethat the stability region becomes smaller. It then converges to the fixed delay
stability region as m → ∞.
We now turn our attention to similarity and dissimilarity of dynamics generated under the fixed time delay to those generated under the continuously
distributed time delay. To this end, we set α = 1, β = 4 and τ = 4 which corresponds to the black point in Figure 7 and Figure 4. Given this configurationof the parameters, complex dynamics under the fixed time delay is obtained
as illustrated in Figure 6 in which many points remain on the vertical dottedline τ = 4 or in Figure 5(C) in which the same dynamics is plotted in the(k(t), k(t − τ )) plane and seems to be very high-order cyclic or aperiodic. In
order to see what dynamics the distributed model can generate under the same
parametric condition, we reduce the integro-differential equation (14) to a system of ordinary differential equations. Since the equivalence is already shownin the case of m = 1, we assume that m > 1. The expected value of capital isgiven by tm(t−σ)
1  m m+1e(t − σ)m e− T k(σ)dσk (t) =T
0 m!
and for all ℓ = 0, 1, ...m, introduce the functions t m(t−σ)
1 m ℓ+1(t − σ)ℓ e− T k(σ)dσxℓ (t) =ℓ!
T0

where xm (t) = ke (t). Then by differentiation,k̇e (t) =ẋℓ (t) =m(xm−1 (t) − ke (t)) for ℓ = m,Tm(xℓ−1 (t) − xℓ (t)) for ℓ = 1, ...m − 1
T

and

m
(k(t) − x0 (t)) for ℓ = 0.
T
Combining these m + 1 equations withẋ0 (t) =k̇(t) = −αk(t) + βke (t)(1 − ke (t))
yields the m + 2 dimensional system of ordinary differential equations for k(t),ke (t) and xℓ (t), ℓ = 0, 1, ..., m − 1. Specifying the value of m, we perform
simulations. When m = 10, the ODE system exhibits a periodic cycle with two
extremes as shown in Figure 8(A). When m increases to 30, a periodic cyclewith six extremes appears, as shown in Figure 8(B). When m becomes morethan 50, erratic behavior may occur and the aperiodic behavior with m = 100
is represented in Figure 8(C).

23Figure 8. Return maps under continuously distributed time delay
Although Figures 8(A), (B) and (C) look similar to Figures 5 (A), (B) and (C),there is a distinctive difference: the values of α, β and τ are fixed and m isaltered in Figure 8 while the value of α and β are fixed and τ is changed in
Figure 5. Notice however the similarity between Figure 5(C) and Figure 8(C),both are obtained under the same values of the parameters α, β and τ andm = 200. So the following result is confirmed in a different way: the distributed
delay model with high value of m generates similar dynamics as the fixed delay
model. To measure the complexity of these dynamics generated by the ODE
system, we calculate the maximum Lyapunov exponent against various valuesof m and the results are plotted in Figure 9 where ML on the ordinate stands
for Maximum Lyapunov:

Figure 9 Maximum Lyapunove exponent againts m
The ODE system has zero maximum Lyapunov exponent for m below 40 and apositive maximum Lyapunov exponent for m greater than 50. That is, chaotic
dynamics are born for some values of m between 40 and 50. The fixed delay
model (14) can give rise to chaos as numerically confirmed in Figure 6. It looks
like a simple equation, however, it is actually infinite dimensional dynamic system. If the distributed delay model with higher value of m can approximate
dynamics generated by the fixed delay model, it can be safe to say that chaos
generated by the fixed delay model is thought to be high dimensional chaos
24since the distributed delay model can generate the similar complex dynamics
only when m takes a large value. With these simulations and the results obtained in Section 3, we summarize the main points of the dynamic model with
continuously distributed time delay:
Theorem 4 Under Assumption 3, dynamics in the neoclassical growth modelwith continuously distributed time delay is the following:(1) it is locally asymptotically stable when the shape parameter m is zero or unity; (2) it generate
periodic cycles for 2 ≤ m ≤ 40; (3) it gives rise to aperiodic fluctuations form ≥ 50; (4) it converges to the neoclassical growth model with fixed time delay
as m goes to infinity.

6

Concluding Remarks

The neoclassical growth model is developed in this paper with continuous timescales and two kinds of time delay. It complements the pioneering work of
Day (1982) in which discrete-time and a production lag are assumed. Special
function form is introduced to consider pollution in production, which resultsin the dynamic model equivalent to that induced by the logistic map. Ourcontinuous-time model is similar in spirit and functional form to Day’s discretetime model.
The capital accumulation process has been formulated in a way that allows
for both continuously distributed and fixed time delays. They are a clear consequence of delays in information gathering in the production process and ininvestment decisions. We start with the continuous model without time delay
and confirm the well-known result that the steady state k∗ is always asymptotically stable if α < β. Our first concern of this paper is on the stability
preservation in the delay models. Theorems 1 and 2 show that the same statek∗ in the delay model is always asymptotically stable, regardless of the lengthof the time delay, if β ≤ 3α. This condition holds if the technology parameter
A is sufficiently small. In addition to this, Case 2 with T > 0 and m = 0 implies
asymptotic stability of k∗ . Thus, our first conclusion is that the delay becomes
harmless to stability of the steady state if the technology parameter takes asmall value or the weighting function is exponentially declining.
Our second concern turns to the destabilizing effect caused by time delay
when β > 3α. In the case of continuously distributed time delay, the capital
accumulation process becomes a set of integro-differential equations and under
specified conditions, local instability may occur. The possibility of the birthof limit cycles is analytically verified based on Hopf bifurcation. The shape ofthe separation curves depends on the value of the shape parameter m of theweighting function. The separation curve takes a fork-shaped profile for a small
number of m indicating that the time lag has to be sufficiently small or sufficiently large to guarantee local asymptotical stability as depicted in Figures 1
and 4. Further, numerical simulations show that some segments of the cyclic
trajectories enter into the non-feasible negative region (Figure 2). This cannot
happen if deviations of time delay from its critical values are small enough asshown in Figure 1 and Figure 3. However this is not the case in large deviations.
In order to force the trajectory to move back to the positive region from zero values, we impose the non-negativity constraint on output in the capital dynamic

25

equation. This is a simple way to prevent trajectories from being negative. In a
more sophisticated way, the mound-shaped production function has to be specified more properly. In our next paper this issue will be further elaborated andthe corresponding dynamic model investigated in which the economic variables
remain non-negative without any nonnegative constraints. Our second conclusion is that the model with continuously distributed time delay exhibits various
dynamics including stability, periodic-cycles and aperiodic-cycles, depending on
the value of m as shown in Figure 8.
Next we examine the destabilizing effect when the dynamic system moves
to fixed delay from distributed delay. We analytically derive the critical valueof the length of the fixed delay at which the stability switch occurs and thennumerically demonstrate that increasing the length of the fixed delay from thecritical value alters movement of the dynamics from stable steady state, throughperiodic cycle, to chaos. The simulation result suggests that longer fixed delay
induces complex behavior involving chaos more likely. Our third conclusion isessentially the same as our second: there is a period-doubling cascade to chaos
if τ increases from its critical value τ ∗ .
An important relation between continuously distributed and fixed time delays has been pointed out in this paper. We have shown that the characteristic
equation of the distributed case converges to that of the fixed delay model when
the shape parameter of the weighting function tends to infinity. This interesting property of the growth model has been also illustrated by the convergenceof the stability regions. This means that when the distributed delay becomes
more narrowed, cyclic and chaotic behavior are more pronounced. The destabilizing effect can be summarized as follows: time delay could change dynamics
significantly implying that the steady state bifurcates to chaos through perioddoubling cascade if the length of τ increases in the fixed delay model while thesame phenomenon occurs if the value of m increases in the distributed delay
model. The main result of this paper is to show the birth of limit cycles andpossible chaotic behavior in the continuous-time neoclassical growth model if
the delay is taken into account.

26

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27