Discussion Paper No.227

Nonlinear Multiplier-Accelerator Model with
Investment and Consumption Delays
Akio Matsumoto
Chuo University

Ferenc Szidarovszky
University of Pécs

June 2014

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Nonlinear Multiplier-Accelerator Model with
Investment and Consumption Delays
Akio Matsumotoyand Ferenc Szidarovszkyz

Abstract
This paper shows how cyclic dynamics of national income can emerge
in the multiplier-accelerator model with continuous time scale when delays
in investment and consumption are presented. An S-shaped functional
form of investment and a linear consumption function are adopted toillustrate the phenomenon and to compute the stability-switching curves
on which a stability gain or loss occurs. Assuming that the equilibrium
national income is locally stable if there are no delays, it is demonstrated
that one delay is harmless and with two delays, the system can produce
limit cycles and the stability switch repeatedly occurs when one of thedelays increases and the other is kept to be positive constant.

Keywords: Nonlinear multiplier accelerator model, Business cycle,
Investment delay, Consumption delay, Stability switch

JEL Classi…cations: C63, E12, E32

The authors highly appreciate the …nancial supports from the MEXT-Supported Program
for the Strategic Research Foundation at Private Universities 2013-207, the Japan Society for
the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 24530202, 25380238 and26380316). The usual disclaimers apply.
y Professor, Department of Economics, Senior Researcher, International Center for further
Development of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan; akiom@tamacc.chuo-u.ac.jp
z Professor, Department of Applied Mathematics, University of Pécs, Ifúság, u. 6., H-7624,
Pécs, Hungary; szidarka@gmail.com

1

1

Introduction

Macroeconomic variables such as national income, employment rate, interest
rate, exchange rate, etc. exhibit persistent and irregular ‡uctuations. Althoughlinear models may be aperiodic if exogenous shocks are appropriately introduced, it has been a main interest in studying macroeconomic dynamics to detect endogenous sources of such irregularity. "Delay" and "nonlinearity" wereonce thought to be two of the main ingredients for endogenous cyclic behavior.
If we look back at the classic literature in the 1930’s –1950’s, we …nd jewelry papers on macro dynamics, Kalecki (1935) assuming a gestation lag of investment,
Samuelson (1939) building a multiplier-accelerator model, Kaldor (1940) adopting nonlinear investment based on the pro…t principle, Hicks (1950) extending Samuelson’s model with the ‡oor and ceiling, Goodwin (1951) assembling
Samuelson’s model in a di¤erent way with a nonlinear delay investment based on
the acceleration principle. However, maybe due to mathematical di¢ culties todeal with delay di¤erential equations or nonlinear di¤erence equations, both ofwhich are main tools for analyzing economic dynamics, macro dynamic studies
with delay and/or nonlinearity gradually disappear from the main stream.
In the early 1980’s, chaos theory sheds light on the roles of nonlinearity indeterministic processes to explain various complex dynamic behaviors of macro
economic variables. A lot of e¤orts, since then, have been devoted to investigate
the applicability of the chaos theory to dynamic economic analysis and to reconsider the various existing economic models. Samuelsonian multiplier-accelerator
models that we draw much attention to in this study are also a¤ected and extended in various directions. The original model is linear and can generate
only damped or explosive oscillations. Westerho¤ (2006) modi…es the induced
investment to depend on a nonlinear mix of extrapolative and regressive expectation formations. Further, Lines and Westerho¤ (2006) use a weighted averageof extrapolative and reverting expectations formations. Direct consequence of
these alternations is that the adjustment process of national income becomes
nonlinear and the birth of complex output ‡uctuations are numerically con…rmed. Concerning the discrete-time Hicks’version, Hommes (1995) focuses on
consumption and investment delays distributed over several periods and showsthat strange-chaotic attractor can occur. The distinguished feature of the Hicks’
version is an introduction of an income ceiling and an investment ‡oor to theSamuelson version. These exogenous factors are endogenized through a capitalformation theory. The ‡oor is related to the capital stock through a depreciation rate in Puu et al. (2005) and the ceiling to the capital stock through theincome-capital ratio in Puu (2007). Both are taken into account simultaneously
in Sushko et al. (2010). It is shown that periodic, quasi periodic and aperiodic
cycles can emerge in these models. Concerning the continuous-time Hicks’version, Goodwin (1951) replaces the piecewise linearity in an investment function
with a smooth nonlinearity. Puu (2000) chooses truncated Taylor expansionsof a nonlinear investment function. Matsumoto and Szidarovszky (2010) invoke
continuously distributed time delays in consumption and investment and makeit likely that the modi…ed model possesses a piecewisely connected limit cycle.
2This paper purposes to reconsider the lost roles of delays for the emergence of
persistent ‡uctuations. To this end, we extend continuous time Goodwin’s nonlinear multiplier-accelerator model by explicitly dealing with investment delay
and consumption delay. Goodwin’s model is augmented with nonlinear accelerator and investment delay. In demonstrating emergence of persistently cyclic
behavior, the role of nonlinearities has been highlighted whereas the role of thedelays have been made implicit. This is mainly because Goodwin’s delay di¤erential equation that describes dynamics of the national income is approximatedin a neighborhood of zero-delay to obtain a second-order di¤erential equation.
As a natural consequence, considerations on delays lie outside of the scope ofthe main discussion. Recently the role of investment delay is discussed in Matsumoto and Szidarovszky (2014a) in which Goodwin’s original model with one
delay in investment is reexamined. In this paper, we further add consumption
delay to it and investigate whether two delays in investment and consumption are responsible for various macro dynamic ‡uctuations. Although this has
the similar structure to the model considered in Matsumoto and Szidarovszky(2014b) in which two delays in investment and consumption under Goodwin
setting is investigated, there is an essential and important di¤erence. The main
tool used there is not applicable to our model and thus we use a completelydi¤erent method developed by Gu et al. (2005) to analyze it. Further, the twomodels have only a small di¤erence in model construction, the results obtained
are very much di¤erent.
The paper is organized as follows. In Section 2, the basic elements of themultiplier-accelerator model are recapitulated. In Section 3 the one delay model
is considered as a benchmark. In Section 4, the two delay model is analyticallyinvestigated and the stability switch is rigorously considered. In Section 5,
some numerical simulations are presented. Section 6 contains some concluding
remarks.

2

Multiplier-Accelerator Model

We recapitulate the main points of the multiplier-accelerator model of businesscycle. Samuelson (1939) constructs a linear model that combines the multiplier
theory with the acceleration principle and explains the cyclic nature of the ups
and downs in business cycle. The model is cast in discrete-time and based on
the fact that national income at time t is the sum of consumption, Ct ; andinduced investment, It (i.e., Yt = Ct + It ). The model has two ingredients. One
is that consumption is a …xed fraction determined by the marginal propensity
to consume, ; of national income with a one-period delay,
Ct = Yt 1 ; 0 <<1and the other is that induced investment is proportional to changes in consumption in period t that are proportional to changes in national income in periodt 1 (i.e., the acceleration principle),
It = (Ct

Ct 1 ) =3(Yt 1

Yt 2 ):

Simple substitutions yield a second-order di¤erence equation of national income,Yt(1 + )Yt 1 +Yt 2 = 0:(1)

With no autonomous expenditure, the zero solution of equation (1) is the equilibrium national income that is determined by the multiplier. It has been demonstrated that depending on the parameter values, various dynamics including
cyclical oscillations can emerge.1 Since equation (1) is linear, oscillations if any
are mainly damped or explosive but can have constant amplitude under very special conditions. Hicks (1950) reformulates Samuelson’s model as nonlinear (i.e.,
piecewise linear) by introducing the investment ‡oor due to the depreciation of
the existing capital stock and the output ceiling due to the full employment levelof output. Hicks’s model is also in discrete time and its possible formulation is
Ct = Yt 1 ;It = max [ (Yt 1

Yt 2 );IL ] ;(2)

Yt = min [Ct + It ; Y c ]where IL denotes the investment ‡oor and Y c is the income ceiling. These
exogenous bounds prevent the unstable ‡uctuates from being in…nitely explosive. Goodwin (1951) casts the discrete-time model in continuous time scaleand replaces the piecewise linear investment function with the smooth nonlinear function. Five di¤erent versions of his model are presented and its second
version is described by the following two-dimensional system:_"Y_ (t) = K(t)(1a)Y (t)(3)_
K(t)= '(Y_ (t))
where '(Y_ ) denotes the induced investment with '0 (Y_ ) > 0 and '00 (Y_ ) 6= 0.
Delay is introduced in the investment function in the third version,_
K(t)= '(Y_ (t))

which is substituted into the …rst equation in (3) to obtain a delay di¤erential
equation of neutral type,"Y_ (t)'(Y_ (t)) + (1)Y (t) = 0:

(4)

Goodwin (1951) does not deal with equation (4) but considers its linear approximation with respect to ;" Y• (t)(t) + [" + (1) ] Y_ (t)'(Y_ (t)) + (1)Y (t) = 0:

The delay di¤erential equation turns to be a second-order nonlinear di¤erential
equation, which is the fourth version of Goodwin’s model. Under the instability
1 For a qualitative analysis, see, for example, Gandolfo (2009).

4

condition, " + (1
) > '0 (0), and the approximation condition that issu¢ ciently small, it is shown that Goodwin’s di¤erential equation can havea cyclic solution. Matsumoto and Szidarovszky (2014a) rigorously investigate
dynamics generated by equation (4) with larger values of and demonstrate that
equation (4) can produce not only smooth cyclic oscillations but also sawtoothoscillations.
We now move one more step forward. A delay in consumption in the discretetime model is explicitly taken into account in a la Goodwinian continuous-timemodel. In particular, we recast the nonlinear discrete-time multiplier-accelerator
model in continuous time scale with the following modi…cations,C(t) = Y (t);I(t) = '(Y_ (t));

Y (t) =

Z t

1
e
0 "

t
"

(5)

E( )d

where E( ) = C( ) + I( ) is the total expenditure, > 0 is the consumption
delay. The last equation indicates that national income lags behind the expenditure and this delay is of exponential form. The basic dynamic structure ofequation (5) is similar to that of Philips (1954) in which three delays in investment, consumption and production are analyzed in continuous-time scale.
Notice that all the delays are continuously distributed and have exponential
forms in Philips’ model whereas consumption and investment delays are …xed
in system (5). Di¤erentiating the last equation in (5) with respect to t andsubstituting delayed consumption and investment into the resultant expression
presents a di¤erential equation with two …xed delays,"Y_ (t)'(Y_ (t)) + Y (t)Y (t) = 0:(6)

This is the dynamic model we will analyze. Similarity to equation (4) is clear.
It preserves the four main features of the discrete-time multiplier-accelerator
model, the multiplier, the acceleration principle, the nonlinear investment function and the delays in consumption and investment.2 Its linearly approximatedversion is"Y_ (t) + Y (t)
Y_ (t
)
Y (t
)=0
(7)
where

0

= ' (0): With the notationa=1; b = and c ="""equation (7) becomes
Y_ (t) + aY (t)bY_ (t)cY (t) = 0:(8)

2 Professor Anna Antonova indicates this form of the consumption delay in the private cor-

respondence. See Matsumoto and Szidarovszky (2014b) for another form of the consumption
delay in which the last two terms, Y (t)Y (t); are replaced with (1
)Y (t
).

5

The corresponding characteristic equation is obtained by substituting an exponential solution, Y (t) = e t u,
+a

b e

ce

= 0:

(9)

Stability of equation (8) can be examined by …nding the locations of the eigenvalues of equation (9). Before proceeding, we make the following assumption
since, under many circumstances, it is natural to suppose that the investment
delay is longer than the consumption delay.
Assumption 1.

>

As a benchmark for the stability analysis, we …rst consider the continuoustime model without delays,"Y_ (t)'(Y_ (t)) + (1)Y (t) = 0(10)which is obtained by substituting the second equation in (3) into the …rst or
taking away andfrom equation (6). The local asymptotical stability ofequation (10) can be examined by linearization around the steady state Y = 0:
"Y_ (t)Y_ (t) + (1)Y (t) = 0:

If " = ; then Y (t) = 0 for all t 0 is a solution, which is uninteresting. It isalso checked that the steady state is locally asymptotically stable if " > andlocally unstable if " < . In the existing literature, local instability is usuallyassumed for cyclic study.3 Since we will consider the delay e¤ects on the stablesteady state, we impose the following condition in which the steady state of thenon-delay equation (10) is locally asymptotically stable:
Assumption 2. " >3

The Single-Delay CaseIn this section we brie‡y examine the single delay e¤ect on the output dynamics,
taking away Assumption 1. Assume …rst that = 0 and > 0; so equation (9)becomes(1 b) + a ce
= 0:
(11)
At = 0 the eigenvalue is (1)=("); so equation (11) is stable since < "and 0 <
< 1. At any stability switch = i!; where we can assume that! > 0; since the conjugate of any eigenvalue is also an eigenvalue. Substituting
it into equation (11) transforms equation (11) toi(1b)! + ac(cos !

i sin !) = 0

3 Matsumoto (2009) assumes local stability of the steady state and exhibits that a Goodwinean multiplier-acclerator model can have multiple limit cycles.

6and then separating the real and imaginary parts gives two equations for two
unknowns ! and asa c cos ! = 0(1
Moving a and (1yieldb)! + c sin ! = 0:b)! to the right hand sides and adding the squared equationsb)2 ! 2 = c2(1

a2 < 0:This inequality means that there is no positive ! satisfying the last equation.
Thus we conclude that no stability switch occurs and the steady state is locallyasymptotically stable for all
0: That is, consumption delay is harmless.
We now examine the opposite case in which > 0 and = 0; so equation(9) becomes+a c b e= 0:(12)
It can be checked that the steady state Y solves equation (8) with = 0 and islocally asymptotically stable for = 0. Assuming that = i! with ! > 0 is asolution of equation (12) and substituting it into equation (12) presentac!

b! sin ! = 0;b! cos ! = 0:

Moving a c and ! to the right hand sides, squaring them and adding them
together give(a c)2 + (1 b2 )! 2 = 0:
Since 1 b2 > 0 by Assumption 2, there is no positive ! solving the above
equation. Thus no stability switch occurs for all > 0 and the investment delay
is also harmless under Assumption 2. Lastly we consider one more special casein which the delays are identical, = . The characteristic equation is+awhich, with(b + c)e=0(13)= !; can be divided into two equations,c cos ! + b! sin ! = ac sin ! + b! cos ! = !

from which we have(1b2 )! 2 = c2

a2 < 0:This inequality implies that there is no positive ! solving the above equation.
Thus the steady state is locally asymptotically stable for all =0: In otherwords, the single delay does not a¤ect asymptotic behavior. However, it mattersin transient behavior.4 To sum up, we have the following:
4 See Matsumoto and Szidarovszky (2014a) for some numerical results in which the delay
generates oscillations, causes slower convergence and makes time trajectories kinked while thetime trajectory with no delay monotonically converges to the zero solution.

7

Theorem 1 Given Assumption 2, the zero solution of one-delay di¤ erential
equation (8) with = 0, = 0 or = is locally asymptotically stable for all> 0, > 0 or = > 0.

4

The Two-Delay Case

Characteristic equation (9) is now investigated with applying Gu’s method (Gu,et al. (2005)) developed to analyze a two delay di¤erential equation.5 Dividing
its both sides by a + and introducing the new functions,a1 ( ) =ba+

and a2 ( ) =ca+simplify equation (9),a( ) = 1 + a1 ( )eSuppose that

+ a2 ( )e

= 0:

(14)

= i! with ! > 0:a1 (i!) =b! 2
a2 + ! 2

ab!
i 2
a + !2(15)a2 (i!) =

ac
c!
+i 2
:
a2 + ! 2
a + !2(16)

andTheir absolute values areja1 (i!)j = pb!
a2 + ! 2and ja2 (i!)j = p

c

(17)

a2 + ! 2

and their arguments arearg(a1 (i!)) = tan 1a+!

and arg(a2 (i!)) =

tan 1

!
:
a

(18)

We can consider the three terms in a( ) as three vectors in the complex planewith the magnitudes 1; ja1 (i!)j and ja2 (i!)j : The solutions of a( ) means that
if we put these vectors head to tail, they form two triangles in two di¤erent ways.
One triangle is illustrated in Figure 1 and the other is obtained by turning it
5 In Matsumoto and Szidarovszky (2014b), the similar two delay model is analysed with anelementary method, which is unapplicable to analyse equation (9) due to the existence of theconstant term, a:8

over with respect to the axis of abscissa.

Figure 1. Triangle formed by 1; ja1 (i!)j andja2 (i!)jIn a triangle consisting of three line segments, the length of the sum of any
two adjacent line segments is not shorter than the length of the remaining line
segment,
1 ja1 (i!)j + ja2 (i!)j ;ja1 (i!)j

1 + ja2 (i!)j ;ja2 (i!)j

1 + ja1 (i!)j :andSubstituting the absolute values in (17) renders these three conditions to thefollowing two conditions,f (!) = (1b2 )! 22bc! + a2

c2

0g(!) = (1b2 )! 2 + 2bc! + a2

c2

0:

and
Both f (!) and g(!) have the same discriminant,D = 4[c2a2 (1b2 )]:In the following we draw attention to the case of D > 0,6 otherwise f (!) > 0
for all ! implying no stability switch. Solving g(!) = 0 gives the solutionspp
c2 a2 (1 b2 )bc + c2 a2 (1 b2 )bc
and ! 2 =
!1 =
1 b2
1 b2
6 Under Assumption 1, D > 0 when

>

p
1

9

( =")2and so does solving f (!) = 0,pbcc2 a2 (1
!3 =
1 b2

b2 )

and ! 4 =

bc +

p

c2
1

a2 (1
b2

b2 )

:

Since both ! 1 and ! 2 are negative and both ! 3 and ! 4 are positive, the two
conditions, f (!) 0 and g(!) 0, are satis…ed when ! is in interval [! 3 ; ! 4 ]:We will next …nd all the pairs of ( ; ) satisfying a(i!) = 0: The internalangles, 1 and 2 ; of the triangle in Figure 1 can be calculated by the law of
cosine as
a2 + (1 + b2 )! 2 c2
1p
(19)
1 (!) = cos
2b! a2 + ! 2
and
2 (!)

Solving two equations for= cos 1a2 + (1 b2 )! 2 + c2
p2c a2 + ! 2

(20)

:

and

arg a1 (i!)e i ! + 2m

1 (!) =

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

2 (!) =

and
yield
=
and

1h
a
tan 1
+
!
!

+ (2m

i

1)

1 (!)

:

i
1h
!
tan 1
+ + (2n 1)
2 (!)
!
a
where arguments de…ned in (18) are used. Then for any ! satisfying the two
conditions, f (!) 0 and g(!) 0, we can …nd the pairs of ( ; ) constructing
the partition curves for ! 3 ! ! 4 :=C1 (m; n) = f 1 (!; m); 1 (!; n)g
where

1h
a
tan 1
+ 2m +
!
!
1h
!
tan 1
+ 2n
(!;
n)
=
1
!
a

1 (!; m) =

and
where

(21)
i

1 (!)

i
2 (!)

(22)

C2 (m; n) = f 2 (!; m); 2 (!; n)g

(23)

i
1h
a
tan 1
+ 2m
(!)
1
!
!
i
1h
1 !
(!;
n)
=
tan
+
2n
+
(!)
2
2
!
a

(24)

2 (!; m) =10

with m; n = 0; 1; 2; ::: Notice that m and n are selected to be nonnegative integers
so that > 0 and > 0:
Notice …rst that f (! 3 ) = 0 and f (! 4 ) = 0. Then at ! = ! k for k = 3; 4;
a2 + (1 + b2 )! 2k c2
a2 + (1 b2 )! 2k + c2pp
= 1 and=1
2
2b! k a2 + ! k
2c a2 + ! 2k

implying thatare

1 (! k ) =2 (! k ) = 0. Then the initial and end points of C1 (m; n)a!k

1 (! k ; m) =

1
tan 1
!k

1 (! k ; n) =

1 h
!k
tan 1
!k
a

+ 2n

2 (! k ; m) =

1
tan 1
!k

a
!k

+ 2m

2 (! k ; n) =

!k
1 h
tan 1
!k
a

+ 2n

and

+ 2m
i

with k = 3 and k = 4; respectively. In the same way, these of C2 (m; n) areandiwith k = 3 and k = 4; respectively. Clearly C1 (m; n) and C2 (m; n) have thesame initial points and the same end points. Figure 2 depicts the connecting
curves for m = 1 and n = 1: The lower red curve is the segment C1 (1; 1) andthe upper blue curve is the segment C2 (1; 1). Both segments start at the same
initial point S = ( S ; S ) and arrive at the same end pointsE = ( E ; E ) as !
increases from ! 3 to ! 4 :7 The connecting curves takes a cigar-shaped pro…le.
These curves are shifted to the right by increasing the values of m and up by
increasing the values of n:
We show that stability is lost when increasing crosses the C2 (m; n) curvefrom the left while stability is gained when it crosses C1 (m; n) from the left. We
discuss the direction of stability switch in which the solution of equation (14)cross the imaginary axis as ( ; ) deviates from the partition curve. We …rst
show the following result.
Theorem 2 The sign of Q( ; ) is negative for ( ; ) on C1 (m; n) and positiveon C2 (m; n) where Q( ; ) is de…ned ashiQ( ; ) = Im a1 (i!)a2 ( i!)ei!( ) :7 More precisely,S=

1 (! 3 ; 1) =2 (! 3 ; 1) andE=

1 (! 4 ; 1) =2 (! 4 ; 1)S=

1 (! 3 ; 1) =2 (! 3 ; 1) andE=

1 (! 4 ; 1) =2 (! 4 ; 1):

and

11

Proof. Direction of stability switch depends on the sign of Q: The term
a1 (i!)a2 ( i!)e!( ) is written as
b! 2

ab!
i 2
a + !2a2 + ! 2ac
a2 + ! 2c!
i 2
a + !2ei!()

which has the same sign as imaginary part of( !

i!)ei!(ia) ( a)= i( 2 + ! 2 ) (cos !(which has the same sign as cos !(arguments are positive, so
1 (!);

2 (!) 2

Since
tan 1

h

0;

2

i

:

a
!
+ tan 1
+
!
a

tan 1

1) =

a
!
+ tan 1
!
a

=

+

2

2

1 (!) +

2 (!);

we then have
!( 1

1) =

2

1 (!) +

2 (!)

3
;
2

2

implying thatcos[!( 11 )] < 0:

Therefore,
R1 : Q( ; ) < 0 for ( ; ) on C1 (m; n):Similarly, for ( 2 ; 2 ) on C2 (m; n);
!( 2

2)

=
=

implying that

tan 1

2

))

): Notice that in (19) and (20), both

Notice also that for ( 1 ; 1 ) on C1 (m; n);!( 1) + i sin !(a!
+ tan 1!
ah
1 (!)
2 (!) 2cos[!( 2

1 (!)

;

2 2

i

2 )] > 0:

Therefore,
R2 : Q( ; ) > 0 for ( ; ) on C2 (m; n):
The results, R1 and R2 ; complete the proof.

12

2 (!)

As in Gu et al. (2005), we call the direction of any segment of the stability
switch curve with increasing ! the positive direction. As we move along the
curve in the positive direction, the region on the left hand side is called theregion on the left which will be denoted as L and the region on the right handside the region on the right denoted as R. We now can state Proposition 6.1 of
Gu et al. (2005) as follows.
Theorem 3 As ( ; ) moves from R to L; a stability loss occurs if Q > 0 and
so does a stability gain if Q < 0.
Notice in Figure 2 that Q( A ; 0 ) > 0 as point A; ( A ; 0 ); is on C2 (1; 1) andQ( B ; 0 ) < 0 as point B; ( B ; 0 ); is on C1 (1; 1): If we increase the value ofalong the dotted horizontal line starting at 0 ; crosses the C2 (1; 1) curve fromR to L at point A and also crosses the C1 (1; 1) curve from R to L at point
B. Thus according to Theorem 3, stability is lost at point A and gained atpoint B: The zero solution of equation (14) loses stability by entering the cigarshaped domain and regains stability by leaving it. As we will see later, thecigar-shaped domains de…ned for any other values of m and n are overlappedeach other, distorting the shape of the instability region. However the basic
principle of the stability switch is not changed.

Figure 2. Partition curve with m = 1
and n = 1

5

Simulations

We perform four simulations to examine global dynamics generated by delay
equation (6) and occurrence of the stability switch on the partition curve. To

13this end we take

= 0:6 and " = 0:8 and use the S-shaped investment function,'(x) = a2

a1 + a2
a1 e x + a2

1

with a1 = 3 and a2 = 1.8 It is checked that'0 (x) > 0; lim '(x) = a1 andx!1lim '(x) =

x! 1

a2

the last two results of which implies that the upper bound of '(x) is three times
larger than the lower bound.9 It is also checked that Assumption 2 is satis…edand the discriminants of functions f (!) and g(!) are positive. We investigate
how the dynamics of equation (6) depends on the delays, and . In the …rst
simulation, we …x = 1 and increase from 1 to 4 along the dotted horizontal
line at = 1 in Figure 3(A). The six cigar-shaped domains are obtained for
m = 0; 1; 2; 3; 4; 5 and n = 1 and the lower parts of these domains are illustrated
as yellow regions. In order to simplify the exposition, the domains constructedby the curves Ci (m; n) for n
2 are omitted. The red and blue boundary
curves are described by C1 (m; 1) and C2 (m; 1), respectively. Remember thatthe domain shifts to the right as the value of m increases. As denoted by black
dots, the horizontal line crosses the boundaries of these domains seven times at
1 ' 0:02;

2 ' 0:11;

3 ' 1:12;

4 ' 1:62;

5 ' 2:21;

6 ' 3:14;

7 ' 3:31:

It is already con…rmed that the zero solution is locally stable along the axesof abscissa and ordinate on which one of the two delays is zero and also along
the positive sloping black curve which is the 45 degree line and = holds.10
According to Theorem 3, stability is lost at 3 ; 5 and 7 while it is gained at
4 and 5 since the followings hold:(i)crosses the blue curve from R to L and Q > 0 on C2 (1; 1); C2 (2; 1) andC2 (3; 1);(ii)crosses the red curve from R to L and Q < 0 on C1 (1; 1) and C1 (2; 1):

The bifurcation diagram in Figure 3(B) plots the local maximum and minimum of the trajectory against and presents the numerical results concerning
the dynamics when the local stability is lost. The value of is selected to begreater than unity due to Assumption 1. It is observed that the equilibrium
point bifurcates to a limit cycle that expands, shrinks and then merges to the
8 Goodwin (1951) uses
= 0:6 and " = 0:5 in his simulations. Since we make Assumption
2, we change the value of " to 0:8 which is larger than ,a1 a2== 0:75:
a1 + a2
9 Goodwin (1951) imposes this asymmetric condition on his investment function.
1 0 Notice that the ratio of the horizontal axes to the vertial axes is appropriately adjusted

in Figure 3(A).

14equilibrium point and the basic pattern of the birth of the cycle, growth andextinction is repeated for even larger values of :(A) Partition curves(B) Bifurcation diagram

Figure 3. Stability switches with=1

In the second simulation, the value of is changed to 2 and is increased from
2 to 5 along the dotted horizontal line in Figure 4(A). The red and blue curvesof C1 (m; n) and C2 (m; n) for m = 0; 1; :::; 8 and n = 1; 2; 3; 4 are illustrated.
The yellow regions are surrounded by the segments of C1 (m; 1) and C2 (m; 1)
and are the same as the yellow regions in Figure 3(A). The green regions are
surrounded by C1 (m; 2) and C2 (m; 2). Notice that the regions shift upward
when n increases to 2 from 1. Similarly, the orange regions and the light blue
regions are surrounded by C1 (m; 3) and C2 (m; 3) and C1 (m; 4) and C2 (m; 4),
respectively. The horizontal dotted line crosses the partition curves many times,
however the stability switch takes place three times only at
1 ' 2; 11;

2 ' 3:75 and3 ' 4:22:

Stability is lost at 1 and 3 since the horizontal line enters the colored regions
from the left to the right while stability is gained at 2 since the line leaves thecolored regions and enters the white region. As we have shown in Section 3, thezero solution is stable when > 0 and along the axis of abscissa, > 0 and= 0 along the axis of ordinate and = > 0 along the 45 degree line denotedas the black upward sloping line. It is observed that the zero solution is still
stable, due to continuity, in a neighborhood of the 45 degree line. Comparing
Figure 3(A) with Figure 4(A), we can see that the stability region (i.e., thewhite region) becomes smaller as the value of the consumption delay becomeslarger. Further the bifurcation diagram in Figure 4(B) indicates that the basic

15

pattern of the cyclic oscillations are more distorted as(A)

increases.

(B)

Figure 4. Stability switches with=2

In the third and fourth simulations, the value of is increased to 5 and 10;
respectively. The divisions of the parameter space ( ; ) becomes very messy andthus are omitted. The corresponding bifurcation diagrams are given in Figure5(A) and 5(B) in which delay equation (6) gives rise to more cyclic dynamics
as the lengths of delays become larger.

(A)

=5

(B)

= 10

Figure 5. Bifurcation diagrams with respect to6Conclusions

An extension of Goodwin’s continuous time scale multiplier-accelerator modelis examined in which delayed consumption and investment are assumed. Con16

ditions are given …rst to the local asymptotical stability of the steady state
without delays, and then three special one-delay cases are investigated. The
local asymptotical stability of these models are proved when the same holds forthe non-delay model. In the two-delay case the stability-switching curves are
determined on which stability gain and loss occur repeatedly when one of thedelays increases while the other kept to be constant. An S-shaped functional
form of investment and a linear consumption function are selected, and the simulation study shows how cyclic dynamics of national income can emerge. In
the existing literature, cyclic dynamics is obtained when the steady state is locally unstable. It is thus worthwhile to emphasize that in this study, the steadystate is locally asymptotically stable under no delays. It would be interesting
to see what dynamics the delay model can generate if its steady state is locallyunstable.

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