COMPLEX DYNAMICS IN A KEYNESIAN GROWTH M

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Metroeconornica 44: 1 (1993)

pp. 043-064

COMPLEX DYNAMICS IN A KEYNESIAN GROWTH MODEL
Marc Jarsulic
University of Notre Dame
(March 1990: revised August 1992)

ABSTRACT
This paper develops a non-linear differential-delay model of the Keynesian system. The
non-linearities derive from a non-linear relationship between capacity utilization and
profitability, interacting with the effect of profitability on rates of investment. Local
stability properties are established analytically, and numerical techniques indicate the
possibility of chaotic behavior for some configurations of parameters. The results suggest
that profitability and growth can interact to produce some of the apparently stochastic
components of growth cycles.

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1. INTRODUCTION

In his classic development of Keynesian theory, Harrod (1939) showed
how an instantaneous multiplier and an investment accelerator together
would generate unstable growth. This theoretical construction later
received an important modification at the hands of Goodwin (1951). By
introducing a non-linear accelerator, which provided floors and ceilings
for rates of capital accumulation, the explosive model was transformed
into a dynamical system capable of producing limit cycles. Ceilings on
rates of output were provided by capacity constraints, and floors by
autonomous components of aggregate demand. Smooth transitions
between upper and lower levels of investment were assured by the
inclusion of a non-instantaneous multiplier.
In both the Harrod and the Goodwin models there is no discussion of
income distribution or finance. Accelerator theory, which focusses
exclusively on capacity utilization, does not require it. However, there is
good reason to consider integrating distribution into the Keynesian
framework. Empirical research on the U.S. non-financial corporate

business sector has established that profitability does not move in

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Street, Suite 501, Cambridge, MA 02142, USA.

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proportion to capacity utilization. Rather, there is a non-linear relationship, with profitability first rising, then declining as utilization increases.
Moreover, there is a substantial contemporary history of empirical work
which has emphasized the importance of past profits in the determination of current investment expenditures.
When representations of these apparent empirical regularities are

included in a Keynesian model, the result is a dynamical system capable
of significant complexity. As is demonstrated in this paper, they can
readily be combined to produce a non-linear differential-difference
equation. By using analytical methods this equation is shown to have
steady-state, limit cycles and locally unstable outcomes. Furthermore,
numerical analysis of simulation data suggests the existence of chaotic
behavior for certain configurations of parameters. The model points to
ways in which the apparently stochastic elements of growth dynamics
may arise endogenously from the interactions of capital accumulation
and profitability. A non-economic consequence of the exercise is to
indicate the existence of a previously unexamined chaotic attractor.
The model complements the work of Day and Schafer (1985, 1987),
who have shown the possibility of chaotic outcomes in discrete, nonlinear aggregate demand models, in which the non-linearities derive
from interest rate effects. It is similar in spirit, though not in form, to
the discrete-time models of Woodford (1989) and Jarsulic (1993).

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2. A DIFFERENTIAL-DELAYGROWTH MODEL

The construction of the model, which represents a closed economy with
no government sector, begins with the relationship between utilization
and profitability for the economy. The work of several empirical
business cycle researchers (Boddy and Crotty, 1975; Weisskopf, 1979;
Hahnel and Sherman, 1982) has established that the profit rate in the
U. S. non-financial corporate business sector rises during the initial
phases of a business cycle expansion, declines during the latter stages of
the expansion and continues to decline during the contraction. In a
recent econometric study of this empirical behavior, Bowles et al. (1989)
use a multivariate regression analysis of the NFCB profit rate to show
that, ceteris paribus, there is a one-humped relationship between the
profit rate and utilization. When other factors are held constant, the
profit rate rises and then falls, as capacity utilization increases. Therefore, since peaks in capacity utilization may be taken as rough measures

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of business cycle peaks, the regressions point to an empirical regularity
underlying business cycles.
There are, of course, many possible interpretations of this regularity.
It is consistent with Goodwin’s (1967) reserve army of labor analysis;
with technical constraints producing decreasing and then increasing unit
costs; and with the “cost of job loss” explanation advanced by Bowles et
al. (1989). No attempt will be made to choose among these explanations.
In order to include this empirical regularity in the model easily, it will
be assumed that capacity output is proportional to the capital stock.
This means that the utilization rate will be proportional to the outputcapital ratio. While this is a simplification of a very complicated body of
research, it is reasonable and productive.

To represent this non-linearity we will use a piece-wise linear function
of the form

II = G(u) =

if u s u*

Here II is the rate of profit, u is the rate of capacity utilization, a , b , c
and u* are positive constants, and b = ( a + c ) u * . The variables Il and
u , as well as all the other variables to be introduced below, are
implicitly functions of time.

The identification of the output-capital ratio with the capacity utilization rate is a useful
simplification. It keeps the dimensionality of the model low, since additional variables
which might also determine capacity, such as labor force, are omitted. However, it seems
reasonable to conjecture that the main dynamical conclusions would remain in a more
general formulation. This is so, because more complex measures of utilization, which are,
as in the case of Bowles et al. (1989), production function-based, allow for utilization to
rise even in cases when output-capital ratios are falling.
There is evidence consistent with the assumption made in the model. Annual data show

that during the 1948-88 period there are eight years in which the NFCB net after-tax
profit rate reaches a local maximum. In four of these cases, the output-capital ratio
continues to increase for at least one year past the profit rate peak. The profit rate peaks
occur in 1952, 1955, 1959, 1965, 1972, 1977, 1981 and 1985. The output-capital ratio peaks
are in 1953, 1955, 1960, 1965, 1972, 1978, 1981; and this ratio is increasing through 1988.
These data suggest that the simplifying assumption of the model is workable.
The output-capital ratio used in these calculations is NFCB gross domestic product
divided by net capital stock. Net capital stock is the sum of net fixed nonresidential capital
stock, end-of-year inventories, and end-of-year demand deposits and currency. The
measure of after-tax profitability is the ratio of net after-tax operating surplus divided by
net capital stock. Net after-tax operating surplus is gross profits before taxes minus
depreciation charges with IVA and CCA, minus corporate tax liabilities, plus net interest
paid.
The data on profits, capital stock, and GDP were kindly provided by Thomas Michl.
These same data sources were used to calculate the estimates of parameters a and c,
reported subsequently in section 4 of the text.

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To obtain rough estimates for the parameters in (l), we can look at
data for the postwar U.S. economy. Parameter a is estimated as the
average share of after-tax NFCB profits in GDP for the business cycle
expansions during the 1948-88 period. It has a value of approximately
. l . The value of parameter c is estimated by dividing the change in the
profit rate by the change in the output-capital ratio during those periods
where the output-capital ratio is increasing and profitability is declining.
This parameter has values ranging between .1 and .5.
In true Keynesian fashion it will be assumed that investment is

determined neither by perfect foresight, nor by calculations using a
known probability distribution of future events. Instead, it is assumed
that investment is determined under a regime of Keynesian uncertainty,
in which the important economic future is not known. As Kalecki (1971,
pp. 105-10) points out, this will produce financial constraints on firms
and make past profitability an important determinant of current decisions to accumulate. Entry of borrowers into financial markets will be
confined in the main to those who already own capital. Moreover, entry
will not guarantee financing since lenders, who do not know how
investments will turn out, will limit lending relative to the value of
existing capital. They may choose to use past profitability as a measure
of management skill. Even without these supply constraints, borrowers
will want to limit debt. For, if they cannot repay their debt, they may
lose their capital and the differential power to which it gives them
access. This theoretical perspective implies that firms will make most
investment expenditures, and that their decisions will be affected by the
availability of finance from profits.2
These theoretical considerations about investment are consistent with
a substantial body of empirical work. There is a long history of including
lagged profitability in macroeconometric investment functions (e.g.
Clark, 1979; Kopke, 1985; Abel and Blanchard, 1986). An interesting

review and discussion of this literature is to be found in Michl (1987). In
addition to the macro-literature, recently micro-level econometrics has
indicated the importance of the impact past profitability has on current
investment (Fazzari et al., 1988; Fazzari and Mott, 1989; Devereux and
Schiantarelli, 1990). Neither the macro- nor micro-literature, however,

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The effects of financial constraint have recently received extensive development in the
“New Keynesian” literature (e.g. Fazzari et al., 1988; Greenwald and Stiglitz, 1988). In
these formulations Keynes’s idea of genuine uncertainty is ignored, and probabilistic
representations of the future are used. Here we stick with Keynes’s point of view,
unfashionable though it currently ir

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provides a consistent indication of the length of the time-lag necessary
for profits to affect investment.
A simple representation of this KeynesianKaleckian position on
investment is given by

where K is the real value of the non-depreciating capital stock,
n(t - 8 ) is the rate of profit 8 units of time ago, and 0 is a positive
constant. The value of 8 reflects implementation lags in the execution of
investment plans. The value of the coefficient u reflects the outcome of
negotiation in the financial markets and the “animal spirits” of firms.
To estimate the value of 0, note that in the U.S. non-financial
corporate business sector for the 1948-88 period, approximately eightysix percent of all investment was internally financed (Mayer, 1990
p. 310). This gives a value of approximately 1.2 for u.
A timeless Keynesian multiplier gives ue = Y / K = g/s, where
g = k / K , s is the marginal propensity to save,3 and Y is NNP. This, in
turn, implies that the equilibrium utilization rate will be determined by
past profits according to

The average postwar U.S. household savings rate of .07 will be used as
an approximate value for s.
Of course no dynamical process operates instantaneously. To allow
for this we posit the adjustment process
Li = E(ue - u ) ,

(4)

where E is a positive constant. This allows for smooth transitions in u ,
even if there should be shifts in, say, G ( u ) . Equations (1)-(4) give a
differential-delay equation
ri = ~ o G ( u (t O))/s -

EU.

(5)

The dynamics of this equation will be examined.
Note that for the parameter estimates which have been given, and

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The effect of income distribution on demand is ignored by the proportional consumption
function. To include it in this model would have generated additional complications, given
the relationship of distribution to utilization. It is somewhat easier to deal with this
inter-relationship in a discrete framework. See Jarsulic (1993) for an example.

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assuming E = 1, the value of Ega/s = 1.7 when II(t - 8) < u*/s; and
Eab/s varies between 1.7 and 8.5 as b varies between .1 and .5. Since
these values are a combination of point estimates and assumption, they
need to be utilized with due acknowledgment of their limitations.
However, they will be taken to indicate the approximate absolute and
relative values of the parameters.

3. LOCAL STABILITY PROPERTIES OF THE MODEL

The conditions for the local stability of differential-delay equations such
as (5) are derived in Hayes (1950) and reinterpreted by Burger (1956).
These conditions are summarized and discussed in Gandolfo (1971,
519-26), May (1980), and Glass and Mackey (1988, 191-93). To
develop the stability conditions, first note that a non-linear equation of
the form 1 = F ( x , x ( t - 8)) can be approximated by the first-order
parts of a Taylor series expansion around an equilibrium value giving

i = AZ + Bz(t

-

8),

(6)

zyx

where A = a F / a x ( t ) l . * , B = a F / a x ( t - 8)l,*, z = x - x*, and x* is the
equilibrium value of F . The eigenvalues of (6) need to be calculated in
order to determine local stability. To do this, assume that there is a
solution to (6) of the form z = eh'/', where A = u + iu, a complex
number. Substituting this expression in (6) gives

A = A8

+ B8e'

(7)

and it is this expression which must be solved for the eigenvalue A.
Burger (1956) has shown that the real parts of A will be negative if,
and only if, one of the following hold:
(a) -1

6

B 8 and A < - B

(b) B 8 < -1 and A < B

(8)

(c) B s A < - B and [arcos(-A/B)/(B2 - A2)1/2]> 8.

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It is clear from the discussion in Hayes (1950) that the eigenvalue of (6)
will have a zero real part, and hence exhibit neutral stability, when the
B 6 A < - B and [arcos(-A/B)/(B2 - A2)'l2] = 6. Furthermore, the
real part of A will be positive and (6) will be unstable if B S A < - B
< 8.
and [arcos(-A/B)/(B2 -

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Equation (5) will have a non-zero equilibrium value when aa/s = 1,
and when u* = ab/(c + s). On the basis of the parameter estimates
made earlier, and since the condition aa/s = 1 would occur only by a
fluke, the second equilibrium seems more plausible. This implies that,
when applying (8) to (5), A = - I and B = --EOC/S. Given the approximate parameter values already calculated, it appears that (8c) will
determine local stability for the problem at hand.
Locally stable and unstable combinations of 8 and - B , when
- I = A = -1 , are graphed in Figure 1. Since larger values of 8 and I BI
both contribute to instability, a trade-off between the two is necessary to
preserve stability (5). As time-lags become longer, the negative slope of
the non-linear part of ( 5 ) needs to become gentler, if the system is not
to produce unstable behavior. When I = 1, a = 1.05, c = .14, a = 1.2,
s = .07, and B = -2.8, Figure 1 suggests that if 8 > 1, there will be
local instability.
For a given value of 8, factors which raise the absolute value of B
will tend to make the equilibrium point of ( 5 ) locally unstable. In
particular, this movement toward instability will be caused by an
increased level of animal spirits or a liberalization of supply in financial
markets, represented by an increase in a; an increased value of the
multiplier, caused by a decrease in s; or an increased downward
pressure on profitability, represented by an increase in c. Thus, the
model ties instability to traditional Keynesian sources. with an additional
role for profitability.

unstable region

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1

1.001

3.251

5.500

7.750

10.000

-B

Figure 1: Stability regions for equation (5)

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4. SIMULATION RESULTS AND GLOBAL DYNAMICS

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To explore the global dynamics of ( 5 ) , the system was simulated under a
variety of parameter values which violate the requirements for local
stability and are economically plausible. Figures 2-4 show a time-series
produced for different values of A and B , with 8 held constant at 2.1.
In these simulations u * , the value of u at which the function G ( u )
changes direction, is set to .5. In Figure 2, the system exhibits limit
cycle behavior. As IBI is increased, the amplitude of the cycle becomes
greater, as illustrated in Figure 3. When IBI is increased further, the
system exhibits apparently aperiodic behavior, as can be seen in Figure
4.
It is worth remarking that the structure of ( 5 ) apparently provides
upper and lower bounds to the movement of u , without explicitly

U

I

I'

urnin= .87
U,,,

=

1.77

"
,

!I

z

/I

0

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100

Time

Figure 2: Simulation of equation (5)

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aa/s = 2.1

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w c / s = 2.0

U

8 = 2.1

I

I

I
I

I

i

1

~

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ii

u,in

/I

= .67

iI

I.

urnax= 1.7:

I

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100

0

Time

Figure 3: Simulation of equation (5)

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building in turning points. Inspection of ( 5 ) would not necessarily lead
one to expect bounded behavior.
While the time-series of Figure 4 has a family resemblance to those
generated by previously identified chaotic attractors, the resemblance
does not prove that ( 5 ) is truly chaotic. The existence of chaotic
attractors has been established analytically for some differential-delay
equations (e.g. an der Heiden and Mackey, 1982; an der Heiden and
Walther, 1983), but unfortunately the results do not apply to equation
( 5 ) and are difficult to apply generally. Therefore the simulation output
is studied by using numerical techniques.
Chaotic dynamical systems are commonly defined as having sensitive
dependence on initial conditions, a dense orbit, and density of periodic

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U

U,i”

u,,

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= .33
=

1.76

0

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100

Time

Figure 4: Simulation of equation (5)

point^.^ The presence of one of these characteristics does not imply the
presence of any of the others. We will focus first on sensitive dependence. The existence of sensitive dependence means that within any
arbitrarily small volume around a point on an attractor, there is a
second point, the trajectory of which will diverge from the trajectory of
the first. A standard test for the existence of sensitive dependence is the
calculation of the Lyapunov exponents of the dynamical system.
The meaning of Lyapunov exponents is easily seen in the case of

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Formally the definition of chaos for maps (Devaney, 1986, 48-50) includes three
elements:

(a) Sensitive dependence. Let f : J - t J be a map from the set J into itself. If there is a
d > 0, such that x E J and for any neighborhood N of x , there is a y E N and an n 2 o
such that I f ” ( x ) - f ” ( y ) / > d , then there is sensitive dependence.
(b) Dense orbit: f : J + J has a dense orbit if for any open U ,V C J , there is an n such
that f ” ( U ) n V # 0.
(c) Density of periodic points: the set of all periodic points in J is dense in J .

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one-dimensional discrete maps. If a one-dimensional map is defined by
x i = F ( x i - 1 ) , then the single exponent for this map is defined
(Rasband, 1990, 18-19) as
m

I\(X,,)

= lim 1/n

C In I D F ( ~ ~ ) I .

i=O

(9)

n-+w

Here D F is the derivative of F evaluated at xi,and x i = F ’ ( x o ) , where
Fi is the composition of F with itself i times. Thus, if on average the
derivative of F is greater than one, indicating that the map is on average
locally hyperbolic, the Lyapunov exponent will be greater than one and
the map is said to exhibit sensitive dependence.
The Lyapunov exponents of an n-dimensional continuous dynamical
system likewise measure the presence of hyperbolic behavior. There will
be n such exponents. While their definition is slighty more complicated
than that of an exponent for a one-dimensional map (Rasband, 1990,
187-95), there is nothing conceptually different. For a dynamical system
to have sensitive dependence, there must be a tendency for neighboring
trajectories to separate in at least one direction. Hence at least one
Lyapunov exponent will be positive. (In the continuous case the
exponent must be positive for separation to occur, but it need not have
an absolute value greater than one.) However, since chaotic systems are
still attractors, trajectories must “fold over” one another if there is not
to be global explosiveness. Thus the Lyapunov exponents of multidimensional chaotic dynamical systems vary in sign, with at least one
being positive and at least one negative.
There are well known techniques (Froeschle, 1984; Wolf et al., 1985)
for calculating the spectrum of Lyapunov exponents when the governing
equations of a dynamical system are ordinary differential equations or
maps. In order to adapt them to a difference-differential system like ( 5 ) ,
it is useful to employ the “linear chain trick” described by MacDonald
(1978, 13-31). This will allow ( 5 ) to be represented by a set of ordinary
differential equations. Rewrite the non-linear system of the form

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zy
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i = F ( x , ~ ( - t 6))

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where H ( u ) = H p ( u ) = [mP''uP/p!]exp(-mu).
That is, the lagged
term in (10) is replaced by a distributed lag on x , which has the density
function given by H . Note that the mean lag of the distribution H is
given by (1 p ) / m and that the distribution narrows as p increases.
Next, let x* = x 2 + p , and define the variables x 2 to xl+p by the
distributed lags

+

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Given (10)-(12), MacDonald shows that the variables x1 . . . x 2 + p will
satisfy the following set of ordinary differential equations
fl

= F(x17

x 2

= m(x1 - x 2 )

x2+p)

(13)

f2+p

= m(xl+p

- x2+p)

+

Since the mean lag of the distribution H is given by (1 p)/m, and the
distribution narrows as p increases, increasing p while holding
(1 + p ) / m constant will cause (13) to approximate (10) more closely.
Note also that the relationship between (10) and (13) implies that a
differential-delay equation can be represented as an infinite set of
ordinary differential equations.
The linearization given in (13) is used to approximate ( 5 ) for purposes
of simulation. That is, ( 5 ) is approximated by

f2+p

= W l + p -

xz+p).

Since increases in p act to decrease the width of the distributed lag,
changes in the behavior of (13) are to be expected. Simulation bears this
out. When p = 30, (13) exhibits periodic behavior (Figure 5 ) for
parameter values EUO/S = 2.1, ECO/S = 2.8, and (1 p)/m = 2.1. When
p rises to 175, with (1 + p ) / m held constant, erratic behavior occurs
(Figure 6). Thus it appears that quicker responses to past profitability,
represented by narrower distributed lags on profitability, induce more
erratic behavior.

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.9

.8
.7
.6

zy

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.5
.4

.3

.2

1

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.1
100

110

120

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130

140

150

160

170

180

190

2OC

Time

Figure 5: Simulation of (14), p

= 30

Linearization (14) is the vehicle for estimating the Lyapunov exponents of ( S ) , using the techniques described in Wolf et al. (1985, 310-12).
However, since the dynamics of (14) change with dimensionality, it is
reasonable to expect changes in exponent estimates as the dimension
increases. Convergence in estimates is desired. The first five exponents
of the Lyapunov spectrum, calculated for increasing values of p with the
mean lag held constant at 2.1, are presented in Table 1. Although the
absolute values change with the value of p , the largest exponent is
consistently positive, and the next four exponents are consistently near
zero or negative in each of the estimates. There appears to be a
tendency to convergence in the estimated values of the exponents.

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1.o

I

.9

I

.8

.7

k

.6

1

I

\

.5
.4
.3
.2
.1
0

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150.

160

180

170

190

200

210

220

230

240

250

zyx
Time

Figure 6: Simulation of (14), p = 175

Table 1
P

125
175
200
225

Lyapuno v exponents

.lo9
.115
.138
.120

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.0007
-.0001
-.007

.003

-.294
-.245
-.256
-.211

-.663
-.644
-.594
-.578

Lyapunov
Dimension

-1.155
-.965
-.902
-.892

2.37
2.47
2.51
2.152

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Although there are technical reasons which argue against the procedure, techniques developed for analyzing empirical data when the
governing equations are unknown were also applied to simulation output
from equation (5).5 These techniques are described in Wolf et al.
(1985). They produce the estimates of the largest Lyapunov exponent of
( 5 ) given in Table 2. Since these estimates are in the same range as
Table 2
Evolution = 5.0
Max
= .025
Min
= .ow1
Delay

LE

1 .o
1.5
2.5
3.0
4.0

.17
.20
.23
.23
.17

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This table presents estimates of the largest Lyapunov exponent (LE) of equation ( 5 ) , using a
10000-iteration simulation, for combinations of
evolution time (Evolution), time delay between
baseline and nearby trajectories (Delay), and minimum and maximum initial distances between
baseline and nearby trajectories (Min and Max,
respectively). The embedding dimension for all
the calculations is three.

Simulation of mixed difference-differential equations is described and implemented in
W. Schaffer et al., Dynamical Software, User’s Manual and Introduction to Chaotic
Dynamical Systems, 2.23-2.31. The techniques developed there may be indicated briefly.
To integrate a differential-delay equation of the form i = Ax + B x ( t - 0) requires
generating values for x ( t - 0) over an interval, although numerical integrators report
discrete values for each step in the integration. This problem is addressed by linear
interpolation between previously calculated values of x . Thus, if the integrator step size is
s, the lag is 0 = n*s, and w is a vector storing previous values of x , the Fortran code is
written for the integrator subroutine as
XO

= w ( t 0 - n*s)

X1

= w(to

-

(n

+ (XI

xlg

= xo

XP

= A*x

-

zyx
zy
zyx
l)*s)

- XI))*(? -

tO)/S

+ B*xlg,

where to is the beginning time of the current integration step, t is the current time, and xp
is the integrator code for i .

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those generated by use of the linear chain trick, we can be fairly
confident about our estimates.
Another marker of chaotic attractors is their dimension. As Grassberger and Proccacia (1983a, 1983b, 1984) and Berge et al. (1984, pp.
112-14) note, chaotic attractors are often, although not always, characterized by a fractal, i.e. non-integer, Hausdorff dimension.6 Equation (5)
was simulated and the dimension was estimated using GrassbergerProcaccia techniques as implemented in the Dynamical Systems software
p a ~ k a g e .The
~ procedures are graphically illustrated in Figure 7. The

zy

The concept of dimension is one that is common enough in the analysis of dynamical
systems. The concept can be illustrated by using the Cantor open middle thirds set. The
Cantor set is constructed by removing first the open middle third from the closed interval
[0, 11; then the open middle thirds from the two remaining closed segments; then the open
middle thirds from the four remaining closed segments; and so on.
We then ask how many line segments of length e are needed to cover the closed
segments in the set. For e = 1, this number N ( e ) = 1; for e = 1/3, N ( e ) = 2; for e = 1/9,
N ( e ) = 4; and so on. In general N ( e ) = 2m when e = (1/3)m.
Then the dimension of this set is defined as

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zyx
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limInN(e)/h(l/e)
rn -+ m.

When e =(1/3)m, this limit is In2/ln3, a non-integer, "fractal" measure. This definition of
dimension extends naturally to n-dimensional spaces by letting N be the number of
hypercubes of side e required to cover all the points on an n-dimensional object.
In their work, Grassberger and Procaccia show that the correlation integral and the
dimension of a chaotic attractor are often related. The correlation integral is defined as

C(k) = l i m x H ( k , Zi,)/(N(N

-

l)),

i,j

N+m

where N is the number of points sampled on an attractor, Zi, is the Euclidean distance
between points i and j , and H is the Heaviside function defined by

They show that the relationship
C ( k )= kU

holds. They further demonstrate that the value of u is a lower bound to the Hausdorff
dimension of an attractor and that, in practice, the value of u is usually, though not
always, a good estimator of the dimension of an attractor.
To implement the estimation procedure one constructs a set of Takens (1981) embeddings of a time-series from a dynamical system. An n-dimensional Takens embedding is

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Embedding Dimension

In ( k )

5 -7

0

Figure 7: The right-hand graph plots the ratio of A In C ( k ) / A In (k) against A In (k), for the
simulation of system (5). C(k) is the correlation integral and k is the distance between points
in the Takens embedding. The upper left graph shows the fitted regression of lnC(k) on
ln(k) for each embedding dimension. The values of In (k) are those falling between the
vertical bars in the right hand graph. The lower left graph plots the estimated values of the
regression slope against the respective embedding dimension. These values are estimates of
the correlation dimension. Numerical results are in Table 3

produced by synthesizing an n-dimensional vector from a one-dimensional vector. That is,
the observations on a variable u will be lagged against one another to produce the
n-dimensional vector y , = ( u ( t , ) , u(t, 4), . . ., u ( t , + [n - llq)), where t, and q are time
indices. According to Takens any values of 4 and n will preserve topological equivalence
between the synthesized dynamical system and the actual dynamical system which
produced the one-dimensional vector. The vectors y , are, therefore, treated as if they are
observations on the behavior of n-dimensional dynamical system.
Correlation integrals are then calculated for Takens embeddings of increasing dimension.
For each embedding, A I n C ( k ) / A l n ( k ) is plotted against In(k). Next one looks for a set
of values of I n ( k ) where A h C ( k ) / A l n ( k ) is stable. For this set of values of I n ( k ) , a
regression of In C ( k ) against In ( k ) is calculated. The slope of the regression is an estimate
of the coefficient v. If the values of v from successively higher Takens embeddings
converge, then the value to which they converge is taken as an estimate of the Hausdorff
dimension.

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regression estimates of dimension derived from this procedure are given
in Table 3. The estimates converge to a value of approximately 2.3. This
is additional evidence that (5) produces a chaotic attractor for the
parameter values considered.
It is also of interest to consider the dimension of ( 5 ) from another
viewpoint. Kaplan and Yorke (1979) have conjectured that the dimension of an attractor is related to the spectrum of Lyapunov exponents by
the equation

zyx

zyxwvu
zyxwvu
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where D L is the Lyapunov dimension; the Ai are the Lyapunov
exponents, in descending order of magnitude; and where j is defined by
the condition
i

i+l

2 Ai > 0 and 2 Ai < 0.

i=l

(16)

i=l

This conjecture is satisfied by some chaotic dynamical systems. In the
case of ( S ) , the Lyapunov dimension D L is calculated in Table 1 for the
values of p used in the simulation of (14). These values are close to the
dimension estimates derived using the Grassberger-Proccacia techniques.
This may be taken as additional evidence that ( 5 ) is a chaotic dynamical
system for the parameter values investigated.
In short, the simulation study of (5) has produced some evidence
which is consistent with chaotic behavior for some parameter values.
Table 3

n

a0

a1

ci

df

zy

In (kO)

In (kl)

~~~

2
3
4
5

1.27
1.31
1.14
.84

1.77
2.18
2.33
2.38

.06
.06
.05
.08

2
2
2
2

-3.5
-3.5
-3.5
-3.5

-2.0
-2.0
-2.0
-2.0

n: embedding dimension; aO: intercept of regression; a l : slope of regression and estimate
of dimension of attractor; ci: 95 percent confidence interval for the regression coefficient
a l ; df: degrees of freedom for the regression coefficient a l ; kO: the minimum value for k
in the scaling region; k l : the maximum value for k in the scaling region. The estimates
were made using a simulation of 5000 iterations.

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However, not all parts of the definition of chaos have been satisfied and
the results which have been produced have not been established
analytically. Therefore conclusions must be tentative.
If ( 5 ) does, indeed, produce a chaotic attractor, it adds to the
inventory of those identified in the differential-delay framework. Other
such attractors have been discussed by Glass and Mackey (1979), May
(1980), an der Heiden (1979), an der Heiden and Mackey (1982), and
an der Heiden and Walther (1983). The example produced by Glass and
Mackey is studied exhaustively through simulation by Farmer (1982).
There is a good review of differential-delay systems, and additional
references, in Glass and Mackey (1988, pp. 57-81).

5 . CONCLUSIONS

The model developed in this paper has a clear connection to earlier
work in the Keynesian tradition. It starts with the simplest multiplier
model, although it acknowledges that multipliers are not instantaneous.
It preserves the Keynesian emphasis on animal spirits, but allows past
profits to affect aggregate demand in the manner indicated by Kalecki.
Non-linearities enter through the relationship between profitability and
utilization. Some rough empiricism shows that these theoretical assumptions are consistent with data on the postwar U.S. macroeconomy.
Using ballpark values for the relevant model parameters, it has been
shown that stability, limit cycles, and possibly chaos are all outcomes of
this dynamical system.
It is useful to consider the relationship of this construction to our
understanding actual economic dynamics. And it is here that the
simplicity of the model is especially helpful. Given a fixed implementation lag for investment, the movement of the dynamics from steadystate, through limit cycles, to apparent aperiodicity can be the result of
increased “animal spirits,” a more rapid decline in profitability at higher
rates of growth, or an increased multiplier linked to declining savings
rates. That is, more explosive demand conditions or more rapid changes
in income distribution destabilize the system, sometimes very dramatically. These results seem quite consistent with Keynesian intuitions.
Perhaps less intuitively, the simulation results suggest some complex
relationships between lag structure and the dynamics of the system.
Longer discrete lags make instability and possibly chaos more likely.
And as the system moves from a discrete lag to a distributed lag, there
are other apparent regularities. When a distributed lag is narrowed, the

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effect is to move the system to more pronounced cyclical and possibly
chaotic behavior. While in the discrete lag case a long response time is
not good for stability, the effects will be mitigated if there is a
continuum of impulses from previous outcomes to current behavior.
It might also be observed that in an economy with features qualitatively similar to those of the model, the configuration of behavioral
parameters determining the dynamics could change significantly at
times. Long term expectations could shift, and the behavior of profits
could alter, as political and social power shifted. With such shifts, an
economy could move in and out of erratic patterns of behavior - much
as market economies seem to move out of “normal” business cycles and
into major crises, and sometimes back again.
For those who are unsatisfied with shock-based explanations of major
economic movements, this and similar models suggest a variety of
competing accounts for apparently stochastic events.
ACKNOWLEDGEMENTS

I would like to thank William Schaffer for many useful suggestions; Frank Connolly for
helpful discussion of differential-delay equations; Milind Saraph for assistance with the
computational work.

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Department of Economics
University of Notre Dame
Notre Dame, IN 46556

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