Dynamic Goodwin’s Business Cycles with Fixedand Continuously Distributed Time Delays∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Abstract
Dynamic extensions of the delayed Goodwin’s business cycle model are
examined. Conditions for the local stability of the stationary state will
be determined in the case of fixed and continuously distributed time lags.
These conditions and the stability regimes will be compared with diﬀerent
types of weighting functions. We will show that the stability regions forcontinuously distributed time delay are always larger than in the case of
fixed time lags and they converge to the stability region with fixed timelag as the variance of the delay converges to zero..
Key words: fixed time delay, continuously distributed time delay,business cycle

1

Introduction

In this paper the business cycle model of Goodwin (1951) will be examined inwhich a time delay is introduced between decision to invest and the corresponding outlays. There is always a time lag due to the decision making process andthe decision implementation. Time delays can be modeled in two diﬀerent ways.
In the first case fixed time delays are considered, and the characteristic equations of the corresponding dynamic equations are the mixtures of polynomialand exponential terms resulting in an infinite spectrum. The stability theoryof such dynamic equations is much more complicated than that of ordinary differential equations without delays. A comprehensive summary of the relevant
results is presented, for example, in Kuang (1993). The alternative approachis to consider continuously distributed time lags, when at each time period t, aweighted average of all data from zero up to t is used in the dynamic equations.
∗ This paper was prepared when the first author visited the Department of Systems andIndustrial Engineering of the University of Arizona. He appreciated its hospitality over his
stay. The usual disclaimer applies.
† Department of Economics, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan.
akiom@tamacc.chuo-u.ac.jp‡ Department of Systems and Industrial Engineering, Tucson, Arizona, 85721-0020, USA.
sidar@sie.arizona.edu

1This method is very realistic, since the length of a time delay is usually uncertain, so if it is considered a random variable, continuously distributed time lags
represent average delayed data. Very often firms do not want to react to suddenchanges, they prefer to react to an average of past information. In the case ofcontinuously distributed time delays with special weighting functions, we havea finite spectrum similarly to models without time delays.
In this paper we will compare the delayed Goodwin’s model with fixed andcontinuously distributed time lags. We will examine local stability of the stationary state and in the case of stability loss the possibility of the birth of limit
cycles will be examined via Hopf bifurcation. In Matsumoto and Szidarovszky(2009), a linearized version of Goodwin’s model with fixed lag is compared withthe one with continuously distributed lag having the exponentially declining
weighting function. It is demonstrated that both models generate the same dynamics when the lags are small and diﬀerent dynamics when they are large. In
this study we focus on more general weighting functions
The paper develops as follows. After the basic model without time delay will
be introduced and solved, fixed time lags will be considered. We will determine
the stability region in the parameter space of the model. Then the case ofcontinuously distributed time delay will be examined first with exponential andgamma type weighting functions. In addition to deriving stability conditions
and determining the stability regions, we will compare these results for diﬀerent
types of weighting functions. In the last section conclusions will be drawn.

2

Goodwin’s Business Cycle Model

This section is divided into three parts. In Section 2.1, we recapitulate thebasic elements of Goodwin’s model. We, then, adopt an explicit treatment of
the investment lag into the model. We will examine fixed time delays in Section
2.2 and continuously distributed time delays in Section 2.3, and we will show
how such a delay aﬀects the characteristics of the cyclic dynamics.

2.1

Basic Model

Goodwin (1951) presents five diﬀerent versions of the nonlinear accelerator
model. The first version assumes a piecewise linear function with three levels of investment, which can be thought as the crudest or simplest version of
the non-linear accelerator. This is a text-book model that can give a simple
exhibition on how nonlinearities give rise to endogenous cycles without relying
on structurally unstable parameters, exogenous shocks, etc. The second version
replaces the piecewise linear investment function with a smooth nonlinear investment function. Although persistent cyclical output oscillations are shown
to exist, the second version includes a unfavorable phenomenon, namely, discontinuous investment jump, which is not realistic in the real economic world.
"In order to come close to reality" (p.11, Goodwin (1951)), an investment lag isintroduced in the third version. However, no analytical considerations are given2

to this version. The existence of a business cycle is confirmed in the fourth
version, which is a linear approximation of the third version with respect to theinvestment lag. And finally, alternation of autonomous expenditure over time
is taken into account in the fifth version.
To find out how nonlinearity can generate endogenous cycles, we review thesecond version of Goodwin’s model, which we call the basic model,⎧⎨ εẏ(t) = k̇(t) − (1 − α)y(t),(1)⎩k̇(t) = ϕ(ẏ(t)).

Here k is the capital stock, y the national income, α the marginal propensity
to consume, which is positive and less than unity, and the reciprocal of ε apositive adjustment coeﬃcient. The dot over variables stands for time diﬀerentiation. The first equation of (1) defines an adjustment process of the national
income. Accordingly, national income rises or falls if investment is larger orsmaller than savings. The second equation, in which ϕ(ẏ(t)) denotes the induced investment, describes an accumulation process of capital stock based on
the acceleration principle. According to this principle, investment depends on
the rate of changes in the national income. A distinctive feature of Goodwin’
model is to introduce a nonlinearity into the investment function in such a waythat the investment is proportional to the change in the national income in theneighborhood of the equilibrium income but becomes inflexible (i.e., less elastic)
for extremely larger or smaller values of the income. This nonlinear acceleration principle is crucial in obtaining endogenous cycles in Goodwin’s model. We
will next retain this nonlinearity assumption and specify its explicit form. On
the other hand, we depart from Goodwin’s non-essential assumption of positive
autonomous expenditure and will work with zero autonomous expenditure forthe sake of simplicity. A direct consequence of this assumption is that an equilibrium solution or a stationary point of the basic model is y(t) = ẏ(t) = 0 for
all t.
Inserting the second equation of (1) into the first one and moving the termson the left hand side to the right gives the single dynamic equation for thenational income y,εẏ(t) − ϕ(ẏ(t)) + (1 − α)y(t) = 0.

(2)This is a nonlinear diﬀerential equation. Although it is one-dimensional, its nonlinearity prevents deriving an explicit form of the solution. In spite of this simple
form, it is possible to detect local dynamics by examining its linearized version
in a neighborhood of the stationary point and global dynamics by performing
numerical simulations.
The linear version of the income dynamic equation (2) isεẏ(t) − ν ẏ(t) + (1 − α)y(t) = 0,(3)

where ν = ϕ0 (0) is the slope of the investment function at the stationary point.
3

This is a first-order ordinary diﬀerential equation. Applying separation of variables gives a complete solution,y(t) = y0 eλt with λ =

1−α
,
ν −ε

(4)

where y0 is an initial condition. The stationary point is locally asymptoticallystable or unstable according to whether the eigenvalue λ is negative or positive.
Since 1−α is the positive marginal propensity to save, the sign of the eigenvalue
depends on whether the numerator is positive or negative. Thus the stationary
point is locally asypmptotically stable if ν < ε and unstable if ν > ε.

2.2

Delay Model with Fixed Time Lags

Due to the fact that in real economy, plans and their realizations need time totake eﬀects, Goodwin (1951) introduces the investment lag, θ, between decisions
to invest and the corresponding outlays. Inserting θ into the investment functionof the basic model yields the third version of his model,εẏ(t) − ϕ(ẏ(t − θ)) + (1 − α)y(t) = 0.

(5)This is a neutral delayed nonlinear diﬀerential equation, which we call the fixed
delay model. Goodwin does not analyze dynamics generated by this fixed delay
model. Furthermore, to the best of our knowledge, no analytical solutions of thedelayed model are available yet. However, it is possible, again, to investigate
dynamics of the delayed model by using linearization for local dynamics andnumerical simulations for global dynamics. Since a cyclic oscillation has beenshown to exist in the basic model, our main concern is to see how the presence of
the investment lag aﬀects characteristics of such a slow-rapid cycle. To this end,
we analytically investigate the stability of the cycle generated in the linearized
model and numerically detect what eﬀects are caused by the lag on cyclicaldynamics.
The fixed delay model is autonomous and its special solution is constant(i.e., y(t) = 0) so that its linearized version takes the form of a linear neutral
autonomous delay diﬀerential equation,εẏ(t) − v ẏ(t − θ) + (1 − α)y(t) = 0.

(6)

It is well known that if the characteristic polynomial of a linear neutral equation
has roots only with negative real parts, then the stationary point is locallyasymptotically stable. The normal procedure for solving this equation is to try
an exponential form of the solution. Substituting y(t) = y0 eλt into (6) andrearranging terms, we obtain the corresponding characteristic equation:ελ − νλe−λθ + (1 − α) = 0.
To check stability, we determine conditions under which all roots of thischaracteristic equation lie in the left or right half of the complex plane. Dividing
4both sides of the characteristic equation by ε and introducing the new variablesν
A = 1−αε and B = − ε , we rewrite the characteristic equation as
λ + A + Bλe−λθ = 0.

(7)Kuang (1993) derives explicit conditions for stability/instability of the n-thorder linear real scalar neutral diﬀerential diﬀerence equation with a single delay.
Since (7) is a special case of the n-th order equation, applying the result of Kuang(1993, Theorem 1.2) implies that the real parts of the solutions of equation (7)are positive for all θ if |B| > 1. Hence we have the following result.
Theorem 1 If ν > ε, then the stationary point of (6) is unstable for all θ > 0.
If v < ε (i.e., |B| < 1), (7) has at most finitely many eigenvalues with positive
real part. The roots of the characteristic equation are functions of the delay. As
the lengths of the delay change, the roots may change their signs from positive
to negative or vise versa so that the stability of the solution may also change.
Such phenomena are often referred to as stability switches. We will next showthat such stability switchings cannot take place in the fixed delayed model.
For the following discussion we assume that v < ε. The case v = ε will betreated later as a critical case. It can be checked that λ = 0 is not a solutionof (7) because substituting λ = 0 yields A = 0 that contradicts A > 0. In thecase of v < ε, Kuang (1993, Theorem 1.4) shows that if the stability switches atθ = θ̄, then (7) must have a pair of pure conjugate imaginary roots with θ = θ̄.
Thus to find the critical value of θ̄, we assume that λ = iω, with ω > 0 being aroot of (7) for θ = θ̄, θ̄ ≥ 0. Substituting λ = iω into (7), we have
A + Bω sin ωθ = 0,and
ω + Bω cos ωθ = 0.
Moving A and ω to the right hand side and adding the squares of the resultant
equations, we obtain
A2 + (1 − B 2 )ω 2 = 0.

Since A > 0 and 1−B 2 > 0 as |B| < 1 is assumed, there is no ω that satisfies theabove equation. In other words, there are no roots of (7) crossing the imaginary
axis when θ increases. Therefore, there are no stability switches for any θ.
In case ε = ν in which |B| = 1, the characteristic equation becomesλ(1 − e−λθ ) + A = 0.

(8)

It is clear that λ = 0 is not a solution of (8) since A > 0. Thus we can assume
that a root of (8) has non-negative real part, λ = u + iv with u ≥ 0 for someθ > 0. From (8), we have(u + A)2 + v 2 = e−2uθ (u2 + v2 ) ≤ (u2 + v2 ),5where the last inequality is due to e−2uθ ≤ 1 for u ≥ 0 and θ > 0. Hence
2uA + A2 ≤ 0,
where the direction of inequality contradicts the assumption that u ≥ 0 and
A > 0. Hence it is impossible for the characteristic equation to have roots with
nonnegative real parts. Therefore, all roots of (8) must have negative real partsfor all θ > 0. Summarizing the above discussions gives the following theorem.
Theorem 2 In case of ν ≤ ε, the the stationary point of (6) is asymptoticallystable for all θ > 0.
Combining Theorems 1 and 2, we can state that the stationary state is
asymptotically stable if ν ≤ ε and unstable if ν > ε. The stationary state is
asymptotically stable on the boundary of the stable region.

2.3

Delay Model with Continuously Distributed Lags

Continuously distributed time delay is an alternative approach to deal with atime lag in investment. If the expected change of national income is denoted byẏ e (t) at time t and is based on the entire history of the actual changes of thenational income from zero to t, then the dynamic system can be written as thesystem of integro-diﬀerence equations,εẏ(t) − ϕ(ẏ e (t)) + (1 − α)y(t) = 0,ẏ e (t) =

Z t
0

(9)
w(t − s, θ, m)ẏ(s)ds,

where the weighting function is⎧ 1 t−s
− θ
⎪
⎪
⎨ θe
w(t − s, θ, m) =³ ´⎪⎪
⎩ 1 m m+1 (t − s)m e− m(t−s)θ
m! θ

if m = 0,
if m ≥ 1.

Here m is a nonnegative integer and θ is a positive real parameter, which is associated with the length of the delay. We call this dynamic system the distributed
delay model.
To examine local dynamics of the above system in the neighborhood of thestationary point, we consider the linearized version,Z tεẏ(t) − νw(t − s, θ, m)ẏ(s)ds + (1 − α)y(t) = 0.
0

Looking for the solution in the usual exponential formy(t) = y0 eλt and ẏ(t) = λy0 eλt ,6substituting them into the linearized version, we obtainZ tw(t − s, θ, m)e−λ(t−s) ds + (1 − α) = 0.
ελ − νλ0Introducing the new variable z = t − s simplifies the integral as
Z t
Z t−λ(t−s)w(t − s, θ, m)eds =w(z, θ, m)e−λz dz.
00

By letting t → ∞ and assuming that Re(λ) + m
θ > 0, we haveZ ∞1 − z −λzdz = (1 + λθ)−1 if m = 0,
e θe
θ
0

and

Z ∞
0

That is,λθ −(m+1)
1 ³ m ´m+1 m − mz −λz
z e θ edz = (1 +if m > 1.
)
m! θ
m
Z ∞

w(z, θ, m)e−λz ds =

0

with

q=

⎧
⎨ 1
⎩

µ
¶−(m+1)λθ
1+q

if m = 0,m if m ≥ 1.

Then the characteristic equation becomes
µ
¶m+1
λθ
(ελ + (1 − α)) 1 +− νλ = 0.
q(10)

If there are no time delays, θ = 0, then the above equation is reduced to thesame characteristic equation as the one we have already derived above. We will
next examine some simple cases in which analytical results can be obtained.
As mentioned in the Introduction, the case of m = 0 is rigorously discussed in
Matsumoto and Szidarovszky (2009), we examine stability in cases with m ≥ 1.
We expand the characteristic equation (10) by using the binomial theorem toobtain,a0 λm+2 + a1 λm+1 + ... + am+1 λ + am+2 = 0,
(11)where the coeﬃcients ai are defined asa0ak= εθm+1 > 0,
=

½µ

m+1
k

¶

mε +

µ

m+1
k−1

¶

(1 − α)θ mk−1 θm+1−k > 0 for k = 1, 2, ...m,am+1= mm {mε + (m + 1)(1 − α)θ − mν} R 0,

am+2

= mm+1 (1 − α) > 0.
7¾

According to the Routh-Hurwitz stability criterion, the necessary and suﬃcientconditions that all roots of the characteristic equation (11) have negative realparts are the following:(1) the coeﬃcients are positive, ak > 0 for k = 1, 2, ...2m + 1,
(2) the principle minors of the Routh-Hurwitz determinant¯¯ are positive,¯¯¯ a1 a0 0 0 ¯¯¯¯¯ a1 a0 0 ¯¯¯¯ a1 a0 ¯¯¯ a3 a2 a1 a0 ¯mmm
¯ > 0, ...
¯
¯
¯
¯
¯
D2 = ¯> 0, D3 = ¯ a3 a2 a0 ¯ > 0, D4 = ¯¯a3 a2 ¯¯ a5 a4 a3 ¯¯ a5 a4 a3 a2 ¯¯ a7 a6 a5 a4 ¯Case 1. m = 1

Substituting m = 1 into (11) yields
a0 λ3 + a1 λ2 + a2 λ + a3 = 0,

(12)

where
a0
a1
a2
a3

=
=
=
=

εθ2 > 0,(2ε + (1 − α)θ)θ > 0,ε + (1 − α)2θ − ν R 0,
1 − α > 0.

It can be seen that the sign of a2 is not determined. In addition to a2 > 0,
the Routh-Hurwitz criterion requires that the following second- and third-order
Routh-Hurwitz determinants are positive,¯¯¯¯¯ a1 a0 0 ¯¯¯¯¯a a0 ¯
D21 = ¯¯ 1
> 0 and D31 = ¯¯ a3 a2 a1 ¯¯ > 0.
¯
a3 a2
¯ 0 0 a3 ¯Since D31 = a3 D21 and a3 = 1 − α > 0, we have¡ ¢¡ ¢sign D31 = sign D21 ,
where

©
ª
D21 = θ 2(ε + (1 − α)θ)2 − (2ε + (1 − α)θ)ν .

(13)

Notice that D21= a1 a2 − a0 a3 > 0 requires that a2 > 0 since all othercoeﬃcients are positive. Therefore the only stability condition is D21 > 0, thatis,2(ε + (1 − α)θ)2ν<.
2ε + (1 − α)θ
Therefore we have an explicit equation for the partition line,ν=2(ε + (1 − α)θ)2.
2ε + (1 − α)θ
8(14)Let g1 (θ) denote the right hand side, then g1 (0) = ε and for θ > 0, bothderivatives g10 (θ) and g100 (θ) are positive. Figure 4 with m = 1 shows the curveof this function, and the stationary state is locally asymptotically stable if thepoint (θ, ν) is under this partition line.
We now return to equation (12) to show the possibility of the birth of alimit cycle with continuously distributed delay by applying the Hopf bifurcation
theorem. According to the theorem, we can establish the existence if the cubic
characteristic equation has a pair of pure imaginary roots and the real part ofthese roots vary with a bifurcation parameter. We select ν as the bifurcation
parameter and then calculate its value at the point for which loss of stability
just occurs. Substituting (14) into (12), we can obtain a factorized expressionof the characteristic equation along the partition line,(2ε + (1 − α)θ + εθλ)(1 − α + (2εθ + (1 − α)θ2 )λ2 ) = 0,
which can be explicitly solved for λ. One of the characteristic roots is real andnegative and the other two are pure imaginary:
λ1

λ2,3

2ε + (1 − α)θ
< 0,
εθ
s
1−α
= ±i
= ±iω.
2εθ + (1 − α)θ2
= −In order to apply the Hopf bifurcation theorem, we need to check whether
the real part of the conjugate complex roots change its sign as the bifurcation
parameter passes through its critical value. Suppose that λ depends on ν, λ(ν),
and then implicit-diﬀerentiation of (12) shows that

Thus

¡ 2 2
¡
¢
¢ dλ
3εθ λ + 2 2εθ + (1 − α)θ2 λ + ε + (1 − α)2θ − ν
= λ.
dν
¸
d(Re λ)
sign
dν
λ=iω
∙

"

= sign Re

µ

dλ
dν

¶−1 #

λ=iω

£
¤
= sign 2(2εθ + (1 − α)θ2 )where we used the facts that the terms with λ are imaginary and the constantterms are real. Therefore we have¯d(Re λ) ¯¯> 0.
dν ¯λ=iωThis implies that the roots cross the imaginary axis at iω from left to right as
ν increases. Therefore the Hopf bifurcation theorem applies, and thus there is
the possibility of the birth of limit cycles around the stationary point. The left9

part of Figure 3 illustrates a limit cycle in a 3D space when the stationary stateis unstable, and the right part shows an orbit approaching the stationary state
when it is stable.1

Figure1. Limit cycle and stable trajectory in 3D system
Case 2. m = 2
The characteristic equation is quartic in λ,a0 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0
with coeﬃcientsa0 = εθ3 > 0,
a1 = (6ε + θ(1 − α))θ2 > 0,
a2 = (2ε + θ(1 − α))6θ > 0,
a3 = 4(2ε + 3(1 − α)θ) − 2ν) R 0,
a4 = 8(1 − α) > 0

1 z(t) and w(t) are defined as follows:z(t) =] t0

andw(t) =t−s1(t − s) e− θ ẏ(s)ds
θ2

] t
0

furthermore ν is selected as

1 − t−se θ ẏ(s)ds,θ2(ε + (1 − α)θ)2+ 0.05 in the unstable case,2ε + (1 − α)θ
and2(ε + (1 − α)θ)2− 0.05 in the stable case.
2ε + (1 − α)θ

10All coeﬃcients are positive except a3 whose sign is not determined. The RouthHurwitz determinants can be defined in the same way as before,¯¯¯¯¯ a1 a0 0 0 ¯¯¯¯¯ a1 a0 0 ¯¯¯¯ a1 a0 ¯¯¯ a3 a2 a1 a0 ¯
2
2
2
¯,
¯
¯
¯
¯
¯
, D3 = ¯ a3 a2 a1 ¯ and D4 = ¯
D2 = ¯¯
a3 a2 ¯¯ 0 a4 a3 ¯
¯ 0 a4 a3 a2 ¯
¯ 0 0 0 a4 ¯

where
D22
D32
D42

ª
©
= 2θ3 32ε2 + 3(1 − α)2 θ2 + ε(18θ(1 − α) + 4ν) > 0,©
ª
= 8θ3 8(2ε + (1 − α)θ)3 − 2(28ε2 + 12(1 − α)εθ + 3(1 − α)2 θ2 )ν − 8εν 2 ,= 8(1 − α)D32 .

Notice that condition D32 = a3 D22 − a4 a21 > 0 requires for a3 to be positive,
since all other quantities are positive. Therefore, the only stability condition is
D32 > 0, that is,©
ªg2 (θ) = 8θ3 8(2ε + (1 − α)θ)3 − 2(28ε2 + 12(1 − α)εθ + 3(1 − α)2 θ2 )ν − 8εν 2 > 0,so we obtained an explicit expression of the partition line8(2ε + (1 − α)θ)3 − 2(28ε2 + 12(1 − α)εθ) + 3(1 − α)2 θ2 )ν − 8εν 2 = 0. (15)Since g2 (θ) is a concave parabola with g2 (0), there is a unique positive root ν(θ),
and the stationary state is locally asymptotically stable if ν < ν(θ), that is, thepoint (θ, ν) is under the partition line. It is shown in Figure 4 with m = 2.
Simple, but lengthy calculation shows that the delay model with m = 1 has alarger stable region that with m = 2 by verifying thatµ¶2(ε + (1 − α)θ)2g2< 0.
2ε + (1 − α)θ
By the same procedure as in the case of m = 1 above, we can show the birth ofa limit cycle in the case of m = 2 as well.
After we repeat the above procedure for the values of m = 1, 2, 3, 4, 5, thefive partition lines with m from 1 to 5 are depicted in Figure 4. It can be seenthat all lines cross the vertical axis for ν = ε and their slopes become smaller asm increases. Notice that the dotted horizontal line is the partition line in thecase of fixed time delay. This implies that the stable region becomes smaller as
the value of m increases and converges to the region defined with the fixed time11

delay when m tends to infinity.

Figure 2. Partition lines

3

Conclusions

This paper examines the dynamic delayed Goodwin’s business cycle model. Sufficient and necessary stability condition was derived first in the case of fixed
lags. The stability conditions and the stability regions depend on the type of
the weighting function in the case of continuously distributed time lags. Thesestability region shrinks as the variance of the delay decreases, and it converges
to the stability region of fixed delays as the variance converges to zero. In thecase of continuously distributed time lags, we also showed that in the case ofstability loss, Hopf bifurcation occurs giving the possibility of the birth of limit
cycles around the stationary state.

12References

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Oligopolies: Stability and Bifurcations, Berlin/New York, Springer-Verlag,
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[2] Goodwin, R., "The Nonlinear Accelerator and the Persistence of Business
Cycles," Econometrica, 19, 1-17, 1951.
[3] Kuang, Y., Delay Diﬀerential Equations with Applications in Population
Dynamics, Boston, Academic Press, 1993.
[4] Lorenz, H-W., Nonlinear Dynamical Economics and Chaotic Motion,
Berlin/New York, Springer-Verlag, 1993.
[5] Lorenz, H-W., "Goodwin’s Nonlinear Accelerator and Chaotic Motion,"
Journal of Economics, 47, 413-418, 1987.
[6] Lorenz, H-W, and H. E. Nusse, "Chaotic Attractors, Chaotic Saddles, andFractal Basin Boundaries: Goodwin’s Nonlinear Accelerator Model Reconsidered," Chaos, Solitons and Fractals, 13, 957-965, 2002.
[7] Matsumoto, A., and F. Szidarovszky, "Delay Diﬀerential Economic Dynamic
Model," Discussion paper 108, The Institute of Economic Research, Chuo
University, 2009.

13