Coexistence of Multiple Business Cycles
in Goodwin’s 1951 Model∗
Akio Matsumoto
Department of Economics
Chuo University

Mami Suzuki
Department of Management
Aichi-Gakusen University

January 15, 2006

Abstract
It is well-known that Goodwin’s 1951 business cycle model gives
rise to cyclical fluctuations when its stationary point is unstable. This
study shows coexistence of multiple cycles when it is stable: the stableregion of a stationary point is bounded by a unstable limit cycle thatis, in turn, surrounded by a stable limit cycle. In this case, the modelis globally stable in a sense that it does not diverge for any disturbances but its dynamics is diﬀerent for diﬀerent disturbances. Thatis, dynamics is stable and returns to the stationary point for small disturbances but unstable and exhibits persistent fluctuations for large
disturbances. This result implies two issues: Goodwin’s mode has therobustness of cyclic fluctuations regardless of local dynamical properties and corridor stability.
Key words: corridor stability, coexistence of multiple limit cycles,
nonlinear delayed diﬀerential equation, hopf bifurcation.

∗

The earlier version of the paper, entitled "Reconsiderations of Goodwin’s Model," waspresented at the Goodwin workshop in EAEPE conference held in Bremen, Germany, on
10-12 November 2005, and Chuo Meeting for Economics of Time and Space (Chuo-Mets)
held in Tokyo, Japan, on 30-31 August 2005. Both authors appreciate financial supports
from Ministry of Education, Culture, Sports, Science and Technology (Grand-in-Aid for
Scientifit Reseach B, 15330037) and Chuo University (Joint Reseach Project 0382). The
paper benefited from comments by Serena Sordi, Peter Flaschel, Reiner Franke and Willi
Semmler. The usual disclaimer applies.

11

Introduction

In this study, we re-examine the dynamics of the business cycle model constructed by Goodwin (1951) and exhibit coexistence of multiple limit cycles
when a stationary point is locally stable. Goodwin develops a nonlinear accelerator business cycle model and shows that it can bring about a stable
limit cycle when the stationary state is locally unstable. This study focuseson the unstable case on which very few attempts have been made. Thus
it aims to complement Goodwin’s dynamical analysis on nonlinear businesscycle.
Goodwin proposes five diﬀerent versions of his business cycle model. The
first version assumes the piecewise linear function with three levels of investment. It is a text-book model that can give a simple exhibition of thevalue non-linearities have for generating endogenous cycles without relying
on structually unstable parameters, exogenous shocks, etc. The piecewise
linear investment function, which can be thought as the crudest or simplestof the non-linear accelerator, is replaced with a smooth nonlinear investment
function in the second version. Cyclical fluctuations of output are shown toexist and be independent of initial conditions. However the second version
includes a unfavorable phenomenon, namely, investments jump discontinuously, which is not observed in the real economic world. "In order to comeclose to reality" (p.11, Goodwin(1951)), the production lag is introduced in
the third version. Although no dynamical considerations are given to thethird version, an existence of a stable limit cycle is examined in the forth
version, which is a linear approximation of the third version with respect
to the production lag. The fifth version is recently reconsidered by Lorenze(1987) as a forced oscillator system in which the emergence of chaotic motion
is demonstrated.
Thus far, a lot of eﬀorts have been devoted to investigate the dynamic
structure of the fourth version. Recently Sasakura (1996) gives an elegant
proof of the stability and uniqueness of Goodwin’s cycle. Thus it has been
confirmed that Goodwin’s model possesses a unique stable limit cycle. Since
all these results are obtained under the condition for which a stationary point
is locally unstable, we can ask a natural question: is cyclic behavior robust
under locally stable circumstances?
The main result of this study is to provide a positive answer to this question. For this purpose, we augment the fourth version by introducing an
nonlinear investment function of arctangent type and demonstrate coexistence of a stable stationary point, a unstable limit cycle, and a stable limit
cycle. The Poincaré-Bendixson theorem is a usual procedure for detecting
a cycle in a two-dimensional continuous dynamical model whose trajectories2are unstable in a vicinity of the stationary point. However, the theorem isrendered inapplicable since we deal with the local stability case. We combine the theorem with the Hopf bifurcation theorem and then characterize
the global dynamics to show coexistence. Coexistence of multiple cycles isalso shown in Kaldor’s business cycle model by Grasman and Wentzel (1994)
and in a Metzlerian inventory cycle model by Matsumoto (1996) whose approach we resume here.
We also turn our attentions to dynamics of the second version with aunstable stationary point to which only limited eﬀorts has been devoted.
Since it is a nonlinear diﬀerential equation, analytical considerations seemfruitless. Yet, we perform numerical simulations to find what eﬀects the
production lag produces on the characteristics (i.e., length and amplitude)of cyclical fluctuations of output.
In what follows, Section 2 overviews three versions from the second to thefourth of Goodwin’s business cycle model and reveals the characteristics ofGoodwin’s cycle. Section 3 points to the new results that the approximatedversion exhibits corridor stability in which it could be stable for small shocks
but unstable and generates multiple limit cycles for large shocks. Section 4
makes concluding remarks.

2

Goodwin Business Cycle Model

This section is divided into three parts. Each of three versions of Goodwin’s
model are reviewed in each subsection. In particular, we recapitulate the basic element of the second version. Assuming an explicit form of the nonlinear
investment function, we numerically simulate the model to see what dynamics it can generate in Section 2.1. Then we introduce the production lag into
the second version to get the third version and perform, again, numericalsimulations to find how the lag aﬀects the characteristics(i.e., the length ofa period and the amplitude) of endogenous cycles in Section 2.2. Finally, wederive the most popular version by expanding the third version with respect
to the lag and reveal the stability condition in Section 2.3.

2.1

Basic Model

The second model, which we call the basic model, is summarized as follows.
 εẏ(t) = k̇(t) − (1 − α)y(t),(1)
k̇(t) = ϕ(ẏ(t)).
3k is capital stock, y income, α the marginal propensity to consume to bepositive and less than unity, and ε the output adjustment coeﬃcient. The
first equation defines an adjustment process of national income in a such a
way that national income rises or falls if and only if investment is greateror less than savings. The second is an adjustment process of capital stock
based on the acceleration principle, according to which investment depends
on the rate of change in national income. On the other hand, we depart from
Goodwin’s non-essential assumption of positive autonomous expenditure andwill work with zero-autonomous expenditure for the reason of simplicity. An
direct consequence 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
into the first and arranging terms gives dynamics of the national income,εẏ(t) − ϕ(ẏ(t)) + (1 − α)y(t) = 0.

(2a)This is a first-order nonlinear diﬀerential equation. Once we specify an explicit form of the investment function and set an initial income, dynamics of
y is accordingly determined.
About the induced investment, ϕ(ẏ(t)), we adapt a non-linear acceleration principle, by which we mean that the investment can be considered to
be proportional to the change in national income in a neighborhood of theequilibrium income but get inflexible (i.e., less elastic) for the extreme large
or small values of income. Goodwin assumes a piecewise linear investment
function in his simulations. Following his spirit, but for the sake of analyticalconvenience, we assume a smooth nonlinear investment function;
Assumption 1. The investment function takes a form of an arctangenttype,ϕ(ẏ(t)) = δ {tanh(ẏ(t) − a) − tanh(−a)} , δ > 0 and a > 0.
This function has a "ceiling" and a "floor" and is asymmetric when a parameter, a, is a non-zero. In what follows, we set a = 1 to shift the function
upward by π4 and rightward so as to make the function pass through theorigin and δ = 12
so that the ceiling is three time higher than the floor as inπ
Goodwin. We assume, in this and the next subsection, the stationary point
to be locally unstable by making
Assumption 2.

dϕ(0)is greater than ε.
dẏ4

To find out what dynamics the basic model can generate, we perform
numerical simulations. An initial point is set at y(0) = 10, and in throughout
numerical examples below, all the parameters are fixed at ε = 0.5 and α =0.6 as in Goodwin. The numerical results are given in Figure 1 in which
an endogenous cycle is illustrated in its right and the corresponding time
path in its left. Along the mirror imaged N-shaped ẏ = 0 locus in theright, the initial point denoted by I0 is displaced slightly upward to point
A, at which investment switches discontinuously from positive to negative.
In consequence the orbit jumps from point A to point B. With negative ẏ(t)at point B, the national income gradually declines from point B to point C.
Once point C is reached, investment, again, switches discontinuously fromnegative to positive. In other word, the orbit jumps again to point D frompoint C, from which the national income glides toward point A, and thenthe process repeats itself. Thus we have a closed orbit constituting a selfsustaining cycle. The points A and C are critical points at which one of thevariables makes a discontinuous jump. The same dynamics is depicted froma diﬀerent view point in the left of Figure 1. There the corresponding time
path form t = 0 to t = 20 is illustrated. It can be seen that the time path
has a kink at the turning point, which is an alternative expression of the
discontinuous jump. If we interpret one time period to be one calendar year
as in Goodwin, five cycles are observed in 20 years in the left. The length of
one business cycle in this simulation is approximately 4 years. The numericalsimulations imply that the basic model can generate an endogenous cycle ofoutput due to the nonlinearity involved in the investment accelerator, andinvestment suﬀers discontinuous jumps. We sum up the results.
Proposition 1 Due to the nonlinear investment accelerator, a slow-rapid
limit cycle is brought about in the basic model.
Insert Figure 1 Here.

2.2

Delayed Model

In order to come closer to reality and to get rid of discontinuous jumps, we
introduce the production lag, θ, between decisions to invest and the corresponding outlays into the basic model. As a result, the third version is²ẏ(t + θ) + (1 − α)y(t + θ) = ϕ(ẏ(t)).

(3)

This is a first-order delayed nonlinear diﬀerential equation, which we call the
delayed model. Since a cycle has been shown to exist, our next concern is5whether such a lag aﬀects characteristics of the slow-rapid cycle observed in
the basic model. To put it more concretely, we will address the questions:what eﬀects the lag causes on the kink, length and amplitude of the cyclicalfluctuations?
Goodwin refers nothing to dynamics generated by the delayed model.
Furthermore, to the best of our knowledge, any analytical solutions of thedelayed model are not successfully constructed yet. However, it is possible toinvestigate dynamics of the model with the aid of numerical simulations. For
this reason, we conduct simulations with various values of the production lag
and then compare the one result with the other. By doing so, we can reveal
what eﬀects the lag can produce.
In Figure 2, we exhibit four time paths from t = 0 to t = 20 associatedwith diﬀerent values of θ in the left and the corresponding limit cycles in
the right.1 For all these simulations, we choose the same initial condition(i.e., y(0) = 10) and the same parameter values as in the last simulations but
the diﬀerent values of the production lags. As can be observed in the left ofeach figure, the length of a one-period gets longer as the production lag getslarger.2 To put it more concretely, we have the following numerical results.
In Figure 2A, θ = 0.125 and there are four cycles within the 20 time-period,so that one cycle is lengthened to 5 years. In Figure 2B, the lag is doubled tobe θ = 0.25 and there are approximately three and a quarter cycles implying
the 6 years cycle. In Figure 2C, θ = 0.5 and there approximately two anda three-quarter cycles, so that the length of the cycle is approximately 7
years. In Figure 2D, θ = 1 and there are a bit-longer-than-two cycles, sothat the length of the cycle is 9 years or so. It is found from these simulation
results with the one of the basic model that the length of the business cycles
changes, roughly speaking, from 4 years to 10 years according as the lengthof production lag gets larger from zero to one. In his simulations, Goodwin
points out that the length of the whole cycle is 10.7 years, the upswing is
prolonged to 6.5 years and the downswing shorted to 4.2 years. Although it
is not clear from these time paths in Figure 2 below, the asymmetry of the
upswing and downswing is also observed in our simulations. When θ = 1,
careful observation reveals that the cycle has a 9-years period, its upswing islengthened to 4.8 years and its downswing to 4.2 years. The similar results
1

Since one unit of time is assumed to equal to one year, "20 years" is long enough to
see the lag-eﬀects on business cycles.
2
As explained in Goodwin (1951, p.15), there are economic justification to set valuesof structural parameters at ² = 0.5, α = 0.6 and θ = 1. It may be unfair to Goodwin tochange values of θ, taking other parameter values equal. We, however, believe to derive
the qualitatively similar results even if we make proper adjustments.

6

are obtainable in other simulations.
Insert Figure 2 HereIn the right of each figure, a limit cycle is illustrated in the phase space inwhich we plot the national income on the vertical axis and its time derivativeon the horizontal axis. Figure 3 juxtaposes these limit cycles. Comparing
these limit cycles, we find that the limit cycle changes its shape from aparallelogram-wise cycle to a vertically longer ellipse as the production lag
increases from zero to unity (that is, the width of the cycle gets smaller andthe length gets longer). Since the peak and bottom of the output-cycle arereached at the point where the time derivative is zero, the amplitude of thecycle is the distance between two points at which the limit cycle crosses thevertical axis. We also find that the amplitude of the cycles grows larger andthe rate of output change becomes less as the production lag gets longer.
These numerical results are summarized in
Proposition 2 In the delayed model, the production lag has the eﬀects tomake the length of the business cycle longer and the amplitude larger.

Insert Figure 3 Here.

2.3

Approximated Model

To investigate the eﬀect of the production lag on the emergence of cyclicaldevelopment of output, Goodwin expands the delayed nonlinear diﬀerential
equation with respect to θ and then gets the following second-order nonlinear
diﬀerential equation as his forth version,²θÿ(t) + [² + (1 − α)θ]ẏ(t) − ϕ(ẏ(t)) + (1 − α)y(t) = 0.

(4)

We call it the approximated model. One way to learn about the movement of
output is to integrate numerically on a computer for any given set of initialconditions on y and ẏ. Our interest in this subsection is not on the specifics
of how y wonders about in the short run but is in understanding whether the
displaced output eventually return to its stationary point or not. Namely,
we raise the question of the stability, which can be resolved without solving
the nonlinear equation.
To this purpose, we introduce a new variable, x(t) = ẏ(t), and then transform the second-order nonlinear diﬀerential equation into a two-dimensional

7

nonlinear system that can be put, after arranging terms, into the form,
1 ẋ(t) ={ϕ(x(t)) − [² + (1 − α)θ]x(t) − (1 − α)y(t)} ,
²θ
(5)


ẏ(t) = x(t).

On the (x, y) plane, the ẏ(t) = 0 locus is identical with the vertical axis
while the ẋ(t) = 0 locus becomes the mirror imaged N-shaped curve andpasses through the origin. Thus the 2D-dynamical system possesses a uniquestationary point, (x, y) = (0, 0), at which both loci cross each other.
We now proceed to stability analysis of the approximated system to giveinsight into the qualitative behavior in a neighborhood of the stationary
point. By developing a local version of the system around the stationary
point, we obtain the Jacobi matrix of the planar system,
 1
1−α 
{ϕ0 (0) − [² + (1 − α)θ]}−²θ  .
J =  ²θ
1

0The characteristic equation is

λ2 −

1−α
k
λ+
= 0,
²θ
²θ

where we set k = ϕ0 (0) −[² +(1 −α)θ]. The characteristic roots are computed
to be


sµ ¶
2

k
1 k
4(1 − α) 
λ1,2 =±.
−
2  ²θ
²θ
²θ It follows that the product of the characteristic roots is positive due to theassumptions imposed on parameters, 0 < α < 1 and (², θ) > 0,λ1 λ2 =

1−α
> 0,
²θ

and thus excludes that the stationary point is a saddle point. It also follows
that the sum of the characteristic root can be either positive or negative
according to whether the slope of the investment function at the stationary
point is larger or smaller than the sum of other parameter values,
λ1 + λ2 =1{ϕ0 (0) − [² + (1 − α)θ]} R 0.
²θ
8

The stationary point is locally stable or unstable according to whether thesum of terms in braces is negative or positive.
Putting this result in a diﬀerent way, we can say that the parameterregion is divided into two subregions, the stable one and the unstable one,by the critical line satisfyingv = [² + (1 − α)θ],(6)

where v = ϕ0 (0). It depends on the value of the discriminant of the characteristic equation whether the local dynamics is oscillatory or monotonic.
The locus along which the discriminant is zero is determined byp(7)v = [² + (1 − α)θ] ± 2 (1 − α)εθ,that distinguishes the parameter region for real roots from the one for complex roots. These two loci divide the parameter region as shown in Figure 4
in which MS and MU mean monotonic stable and unstable while OS and OU
mean oscillatory stable and unstable. There, we plot θ on the horizontal axis
and v on the vertical axis. For combinations of parameters, θ and v, in either
the light-gray region or the dark-gray region, the characteristic roots are complex and thus the system produces fluctuations. Moreover, the fluctuations
are explosive for the combinations in the light-gray region and damped forthe one in the dark-gray region. On the other hand, for the combinations of θand v in the white regions, the characteristic roots are real and thus the system produces monotonic dynamics that is convergent or divergent according
to whether the combination is in the lower-white region or the upper-white
region. Figure 4 also implies that the approximated model may produce
qualitatively the same dynamics as v increases irrespective of a choice of the
production lag. Namely, given θ, it generates stable monotonic dynamics,
stable oscillatory dynamics, unstable oscillatory dynamics and then stable
monotonic dynamics as v increases from zero.
Insert Figure 4 HereThus far, there have been a lot of studies that focus on cyclic dynamics ofthe approximated model when the stationary point is locally unstable, thatis, the parameters are selected from the region above the critical line (6) of
Figure 4. On the other hand, little attention has been given to the case inwhich the stationary point is stable. In consequence, little is known about
dynamics for the parameter configurations selected from the region below
the critical line (6). We move one step forward and address the following
question: what can be said about the global dynamics of the approximatedmodel when the stationary point is locally stable?
93

Coexistence of Multiple Cycles

To answer the question we raise just above, we first assume that the fixed
point is locally stable and then demonstrate coexistence of a stable stationary point, a unstable limit cycle and a stable limit cycle. In what follows, weselect a slope of the investment function evaluated at the stationary point
as a bifurcation parameter and concern a possibility of the supercritical bifurcation in Section 3.1. With use of an unstable limit cycle to be shown toexist, we construct an invariant set in the state space and apply the PoincaréBendixson theorem to find a stable limit cycle that encloses the unstable limit
cycle in Section 3.2.

3.1

Hopf Cycle

We investigate, with the help of Hopf bifurcation theorem, whether there
exists an unstable limit cycle in the approximated model. According to
the theorem, the Hopf bifurcation occurs if the complex conjugate roots
cross the imaginary axis. Apparently, the characteristic roots are complex
conjugate with zero real part if k = 0. As there are no other roots in twodimensional system, the consideration of the existence of closed orbits iscomplete if the eigenvalues cross the imaginary axis with non-zero speed
at the bifurcation point. Though there may exist several possibilities toparametrize the approximated model, it seems interesting to choose the slopeof the investment function evaluated at the stationary point as the bifurcation
parameter.
Since we assume the local stability (i.e., ϕ0 (0) < ² + (1 − α)θ where α,² and θ are given) in this section, it can be directly seen that there exists avalue v0 = ϕ0 (0) for which
v0 − [² + (1 − α)θ] = 0,
implying that the complex conjugate cross the imaginary axis. As for v > v0 ,respectively v < v0 , the real part becomes positive, respectively negative.
Hence, v0 is indeed a bifurcation value of the approximated model. According
to the Hopf bifurcation theorem, these establishes the existence of closed
orbits in a neighborhood of the stationary point (0, 0) at v = v0 .3
The Hopf theorem, however, is somewhat ambiguous about the nature of
the closed orbit. There are two possibilities, one is that orbits spiral outward
from the stationary point toward a stable limit cycle, namely, the subcritical3

See Lorenz (1993) for the Hopf bifurcation theorem, the stability index and the following coordinate transformations..

10

bifurcation, and the other is that all orbits starting inside the cycle spiral in
toward the stationary point, namely, the supercritical bifurcation. To makethe distinction between the sub- and super-critical Hopf bifurcation, we checkthe stability of the cycle by calculating the stability index.
The approximated model can be written as
¶
µ ¶ µ
µ ¶
f (x)
x
ẋ
+
=J
0
y
ẏ
where J is the Jacobian defined above, and f (x) is a nonlinear term that canbe derived as
1
f (x) =
{ϕ(x) − ϕ0 (0)x} .
εθ
In order to transform the approximated model into the normal form, we makethe coordinate transformation,


0
1
µ ¶
µ ¶


x
u
.
r
=D
with D = 


y
v
εθ
−
0
1−α
Since the matrix D transforms the coordinate system (x, y) into a new coordinate system (u, v), the approximated model is put into
r


1−α
¶
µ ¶ µ
µ ¶
0
−

 u
0
u̇
εθ

r
=
 1−α
 v + f (v)v̇
0εθ

It is well-known that the stability of the emerging cycle depends on up to
third-order derivatives of the nonlinear function, f (v). In the approximatedmodel, the stability index is calculated as
I=

1 000
ϕ (0) > 0.
16εθThe direction of the inequality indicates that the emerging cycle encloses
the stable stationary point and is repelling (i.e., unstable). It follows thata subcritical Hopf bifurcation occurs for v < v0 in the approximated model.
We sum up the result obtained.
Proposition 3 Given the stability of the stationary point, there exists an
unstable limit cycle that encloses a stable equilibrium in the approximatedmodel.
11

3.2

Stable and Unstable Limit Cycles

Taking v to be less than v0 for which the stationary point is stable, weconcern a number of a limit cycle in this subsection. As shown above, aunstable limit cycle exists for v < v0 due to the Hopf theorem. We proceed
to show an existence of a stable limit cycle that encloses the unstable Hopf
cycle when the stationary point is locally stable.
We construct an invariant set in such a way that once an orbit enters theset, it cannot later escape at some future time and then apply the PoincaréBendixon theorem to examine whether a stable cycle can arise in the set. A
typical application of the theorem is to the case in which there is a single
unstable equilibrium in some invariant set. Remove a neighborhood of thestationary point in which all orbits move away from it. Then the theorem
ensures that the remaining set must contain a limit cycle.4 Since the stationary point is now assumed to be stable in the approximated model, we need
another considerations for searching for such an invariant set.
In Figure 5, we may find a point A so high on the vertical axis that an orbit
starting there comes back to meet the vertical axis at a lower point E after
crossing not only the horizontal axis at points B and D but also the verticalaxis at point C. Consider a region bounded by ABCDEA and an open
region bounded by the unstable limit cycle that is shown to exist in Section
3.1. Then the invariant set to be considered can be constructed by deletingthe latter region from the former region. The result is shown in Figure 5 inwhich the shaded region represents the invariant set. The boundary of theinner white region corresponds to the unstable limit cycle that srrounds thestable stationary point. The shaded region thus has no stationary point, andany orbit starting inside the region stays within its inside. Two orbits, one
starting at point a and the other at point b, are examples that are shown
to remain inside of the invariant set and converge to the same limit cycle,
illustrated as a bold curve. On the other hand, it can be seen that a dotted
orbit starting at point c located inside of the white region spirals toward thestationary point. Formally, the Poincaré-Bendixson theorem guarantees anexistence of one stable cycle in the shaded region. Summing up the resultswith Proposition 3 yields the following results.
Proposition 4 Given the stability of the stationary point, a stable limit cycle
coexists with an unstable limit cycle that encloses a stable equilibrium pointin the approximated model.
Insert Figure 5 Here4

See Chaper 2.2 of Lorenze (1993) for an example.

12

In Figure 6, we present a bifurcation diagram in which the amplitudeof the cycle is on the vertical axis and v as bifurcation parameter on thehorizontal axis. vu is the critical value for which the approximated model
loses its stability, and its stationary point is replaced by a limit cycle. Forv > vu , the model is destabilized for small perturbations so that any orbit
moves away from the stationary point. Nonlinearity of the model prevents
it from diverging globally but leads to a unique stable limit cycle. This isessentially the same cycle as the one that Goodwin demonstrates by applying
the Lienard method. On the other hand, for v < vu , it is locally stabilized
but generates an unstable cycle as well as a stable cycle for v in the interval[vs , vu ]. It can be seen that as v decreases from vu , the amplitude of the innerunstable cycles increases and the one of the outer stable cycle decreases. vsis the other critical value for which two cycles coincide each other. For alower v < vs , limit cycles no longer exit.
Coexistence of multiple cycles and a stable stationary point reminds us"corridor stability," the notion of which was introduced by Leijonhufvud[1973]. It implies that a dynamical system is stable for small perturbations
but unstable for large perturbations. The inside of white circle region in
Figure 5 is the corridor in which the stationary point is stable. Thus, if
perturbations around the stationary point are small enough not to take orbits
out of the corridor, the restoring eﬀect of the model works to return tothe stationary point. To the contrary, if perturbations are large enoughto take orbits outside the corridor, the lasting eﬀect of the model leads topersistent cyclic fluctuations. The foregoing numerical analysis indicates that
non-linearity of investment function can be a source of corridor stability in
Goodwin’s approximated model.
Insert Figure 6 Here

4

Concluding Remarks

We investigate the characteristics and the robustness of Goodwin’s businesscycle in this study. It is confirmed numerically that the nonlinearity of theinvestment function generates cyclical fluctuations of output as shown in Figure 1 and summarized in Proposition 1. It is also confirmed numerically thatthe longer production lag makes the amplitude of the cycle larger and thelength of its period longer as shown in Figure 2 and summarized in Proposition 2. Further, it is seen in Figure 6 that the amplitude of the cycle issensitive to the nonlinearity of investment function and gets larger as thenonlinearity gets stronger. It is demonstrated analytically that cyclic fluctuations of output are brought about even when the stationary state is locally13stable as summarized in Propositions 2 and 3. Coexistence of multiple cycles
and the stable stationary point implies the corridor stability of Goodwin’s
business cycle. This study complements the existing studies on Goodwin’s
business cycles, the studies on which focus mainly on circumstances in which
the stationary state is locally unstable.

References[1] Goodwin, R., "The Nonlinear Accelerator and the Persistence of Business
Cycles," Econometrica, vol.19, 1-17, 1951.
[2] Grasman, J., and J. Wentzel, "Co-existence of a limit cycle and an equilibrium in Kaldor’s Business Cycle Model and its Consequences," Journalof Economic Behavior and Organization, vol.24, 369-377,1994.
[3] Leijonhufvud, A., "Eﬀective Demand Failures," Swedish Journal of Economics, 27-48, 1973.
[4] Lorenz, H-W., Nonlinear Dynamical Economics and Chaotic Motion,
Springer-Verlag, 1993.
[5] Lorenz, H-W., "Goodwin’s Nonlinear Accelerator and Chaotic Motion,"
Journal of Economics, vol. 47, 413-418, 1987.
[6] Matsumoto, A., "A Non-linear Macro Model of Endogenous Inventory
Oscillations," Research in Economics, vol.51, 101-129, 1997.
[7] Puu, T., Attractors, Bifurcations, and Chaos, Second Edition, SpringerVerlag, 2003.
[8] Sasakura, K., "The Business Cycle Model with a Unique Stable Limit
Cycle," Journal of Economic Dynamics and Control, vol.20, 1763-1773,
1996.

14

yHtL

yHtL

A

B

yH
I0

yL

I0

5

10

15

20

y° HtLt
C

Figure 1. Endogeneous Cycle with Discontinuous Jumps.

15

D

yHtL

yHtL

y° HtL

t
5

10

15

20

Figure 2A. Cyclical Fluctuations with θ = 0.125

yHtL

yHt L

y° HtL

t
5

10

15

20

Figure 2B. Cyclical Fluctuations with θ = 0.25.

16

yHtL

yHtL

y° HtL

t
5

10

15

20

Figure 2C. Cyclical Fluctuations with θ = 0.5.

yHtL

yHtL

y° HtL

t
5

10

15

20

Figure 2D. Cyclical Fluctuations with θ = 1.

17

yHtL

q=0

q=1

q=0.5

q=0.25q=0.125y° HtL

Figure 3. Juxtaposition of limit lycles with diﬀerent lag.

18

v

@MU D
@OUD

@OSD

@MS D

q

Figure 4. Division of the Parameter Region.

19

yHtL
A

a
E
b
c

B

D
C

Figure 5. Coexistence of Limit Cycles.

20

xH t L

y

v
vs

vu

Figure 6. Bifurcation Diagrame.

21