Download Oscillatory dynamics of GDP components

Oscillatory dynamics of GDP components
Ionut Purica1
Abstract
GDP and its components’ evolution show an oscillatory behavior. As an alternative
approach to classical cyclic behavior producing models, the paper analyses this behavior
both by Fourier transforms of the data series and by a discussion of the roots
configuration of the associated second order differential equations describing the process.
Specific cycles are identified that associate with the economic sectors contributing to the
generation of GDP. This oscillatory system based approach is providing a complementary
way to describe economic behavior dynamics.
Keywords: nonlinear models, oscillatory behavior, GDP cycles
JEL Classification: C3, C61, C62, D7, D87
Aknowledgements: This paper is suported by the Sectorial Operational Programme Human
Resources Development (SOP HRD), financed from the European Social Fund and by the
Romanian Government under the contract number SOP HRD/89/1.5/S/62988.
Introduction
Let us introduce the reader into the way cyclic behavior has started to be acknowledged and
approached in economics mostly in the first half of the XX-th Century and give a few examples (a
synthesis from the net) of the attempts made by reputable scientists to model such behavior.
Let’s start with a few relevant quotes:
"The general character and agreement in the periodic turn in movements of factors of circulation -these are the specific problems of business cycle theory which have to be solved within the closed
interdependent system....If a business cycle theory which is system-conforming cannot be built, then
"general overproduction" will not only drive the economy but also economic theory into a crisis."
(Adolph Lowe, "How is Business Cyle Theory Possible at All?", WWA, 1926: p.175).
"Since we claim to have shown in the preceding chapters what determines the volume of
employment at any time, it follows, if we are right, that our theory must be capable of explaining
the phenomena of the Trade Cycle."
(John Maynard Keynes, General Theory, 1936: p.313)
"Keynesian economics, in spite of all that it has done for our understanding of business fluctuations,
has beyond all doubt left at least one major thing quite unexplained; and that thing is nothing less
than the business cycle itself....For Keynes did not show us, and did not attempt to show us, save by
1
IPE-INCE-Romanian Academy, [email protected]
a few hints, why it is that in the past the level of activity has fluctuated according to so definite a
pattern."
(John Hicks, Contribution to the Theory of the Trade Cycle, 1950: p.1)
Roy F. Harrod laid out the basic tenets of the Oxbridge research programme in his 1936 The Trade
Cycle: An essay -- much of it written before Harrod even saw a draft of J.M. Keynes's General
Theory. Harrod contended "that by a study of the interconnexions between the Multiplier and the
Relation the secret of the trade cycle may be revealed" (Harrod, 1936: p.70). This "Relation" was
the acceleration principle of investment.
One ought to note that J.M. Keynes himself did not have much credence in a deterministic
accelerator as employed by Harrod and the Oxbridge models. Instead, Keynes had argued that it
was expectations dynamics that generated cycles by affecting the marginal efficiency of investment
and subsequently the multiplier and output (Keynes, 1936: Ch.22). Nevertheless, Keynes left the
topic undetailed. Thus, Roy Harrod went on alone, in his theory of the trade cycle (1936) and later
on in his theory of growth (1939, 1948), to explore the relationships between the Keynesian
multiplier and accelerator-type investment functions to explain a growing, progressive economy
with and without cycles.
The principle of the multiplier, as laid out by R.F. Kahn (1931) and J.M. Keynes (1936) is that if
investment increases, there will be an increase in output as a result of a "multiplier" relationship
between equilibrium output and the autonomous components of spending, in this case:
Y = I/(1-c)
where c is the marginal propensity to consume, Y is output and I is investment. The principle of the
accelerator, as laid out by Albert Aftalion (1913) and John Maurice Clark (1917), was that
investment decisions on the part of firms are at least in part dependent upon expectations of future
increases in demand, which may, in turn, be extrapolated from any current or past increases in
aggregate demand or output, e.g.
It =  (Yt - Yt-1)
Thus, the multiplier principle implies that investment increases output whereas the acceleration
principle implies that increases in output will themselves induce increases in investment.
Consequently, it would at least seem natural if some bright economist put these two together and
examined the dynamic properties of investment and output as they affect each other, perhaps in
generating cycles and/or growth. The first such bright economist was Roy F. Harrod (1936), albeit
his analysis was purely verbal and not without some knots. His associated attempt to formalize a
Keynesian growth model (Harrod's, 1939, 1948) was not much more successful: he ended up with
his famous "knife-edge" instability.
Harrod's fellow Oxford economist, John Hicks (1949, 1950) picked up where Harrod left off.
Hicks's (1950) trade cycle model sought to recast Harrod's unstable "multiplier-accelerator"
dynamics into cyclical ones by having explosive trajectories bang up against floors and ceilings. To
this end, Hicks employed the formalism of dynamical difference equations, that had been
introduced in a similar context by Paul Samuelson (1939) in his income-expenditure "oscillator"
and by Lloyd Metzler (1941) in his inventory cycle.
Hicks's (1950) "forced non-linearity" of ceilings and floors were somewhat restrictive. The
essential Hicksian model was expanded upon by James Duesenberry (1949) and Arthur Smithies
(1957) to include "ratchet effects" and thus obtain cycles along a output growth path. Later,
Duesenberry (1958) and Luigi Pasinetti (1960) considered a different accelerator for the Hicksian
model that would yield both growth and cycles. A bit more distinctly, Richard Goodwin's (1951)
exercise added a non-linear accelerator to generate cycles endogenously while D.J. Smyth (1963)
attempted to incorporate a monetary "LM" side to the standard multiplier-accelerator model. We
shall review these "simple" multiplier-accelerator models before turning to endogenous cycle
theory, where "natural" non-linearities are put in place to generate cycles and growth.
Paul Samuelson credits Alvin Hansen rather than Harrod for the inspiration behind his seminal 1939
contribution. The original Samuelson multiplier-accelerator model (or, as he belatedly baptised it,
the "Hansen-Samuelson" model) relies on a multiplier mechanism which is based on a simple
Keynesian consumption function with a Robertsonian lag:
Ct = c0 + cYt-1
so present consumption is a function of past income (with c as the marginal propensity to consume).
Investment, in turn, is assumed to be composed of three parts:
It = I0 + I(r) +  (Ct - Ct-1)
The first part is autonomous investment, the second is investment induced by interest rates and the
final part is investment induced by changes in consumption demand (the "acceleration" principle). It
is assumed that 0 <  . As we are concentrating on the income-expenditure side, let us assume Ir = 0
(or alternatively, constant interest), so that:
It = I0 +  (Ct - Ct-1)
Now, assuming away government and foreign sector, aggregate demand at time t is:
Ytd = Ct + It = c0 + I0 + cYt-1 +  (Ct - Ct-1)
assuming goods market equilibrium (so Yt = Ytd), then in equilibrium:
Yt = c0 + I0 + cYt-1 +  (Ct - Ct-1)
But we know the values of Ct and Ct-1 are merely Ct = c0 + cYt-1 and Ct-1 = c0 + cYt-2 respectively,
then substituting these in:
Yt = c0 + I0 + cYt-1 +  (c0 + cYt-1 - c0 - cYt-2)
or, rearranging and rewriting as a second order linear difference equation:
Yt - (1 +  )cYt-1 +  cYt-2 = (c0 + I0)
The solution to this system then becomes elementary. The equilibrium level of Y (call it Y p, the
particular solution) is easily solved by letting Yt = Yt-1 = Yt-2 = Yp, or:
(1 - c -  c +  c)Yp = (c0 + I0)
so:
Yp = (c0 + I0)/(1-c)
The complementary function, Yc is also easy to determine. Namely, we know that it will have the
form Yc = A1r1t + A2r2t where A1 and A2 are arbitrary constants to be defined and where r1 and r2 are
the two eigenvalues (characteristic roots) of the following characteristic equation:
r2 - (1+ )cr +  c = 0
Thus, the entire solution is written as Y = Yc + Yp is:
Y = A1r1t + A2r2t + (c0 + I0)/(1-c)
where the constants, A1 and A2 are solved by allowing for particular values of Y0 and Y1 (initial
conditions).
The roots determine the character of the dynamics of the system. The solution of the characteristic
equation is easily determined by the elementary quadratic equation:
r1, r2 = [(1+ )c   [((1+ )c)2 - 4 c)]]/2
from where we can therefore derive the parameter regimes of  and c which yield the different
dynamics. Firstly, we must establish the ranges of  and c which yield complex as opposed to
asymptotic dynamics. This is observed in the discriminant D = ((1+ )c)2 - 4 c. Now, if D  0, then
we have regular dynamics; if D < 0, we have complex dynamics. Thus, for regular dynamics, it
must be that ((1+ )c)2  4 c or simply:
c  4 /(1+ )2
In Figure 1 below, we have drawn the function c = 4 /(1+ )2 in (c,  ) space. It is obviously a nonlinear function bounded from above by c = 1 (note that when  = 1, then c = 4/22 = 1; any other
value of  will give 0 < c < 1)). So, in the area below the curve c < 4 /(1+ )2 and we have
complex dynamics (or oscillations), whereas above the curve, where c > 4 /(1+ )2 we have
regular dynamics.
Let us now assume that D > 0. Now, by the Schur Criterion, we know that sufficient conditions for
damped monotonic behavior are:
(i) 1 - (1+ )c +  c  0
(ii) 1 -  c > 0
Or, rewriting both as 1 - (1+ )c  -  c and -  c > -1, and consequently combining them, 1 - (1+
)c > -1, the Schur conditions reduce to:
2 > (1+ )c
as a sufficient condition for damped, monotonic dynamics. Now, taking our discriminant, D = ((1+
)c)2 - 4 c, we know that for D > 0, it must be that:
(1+ )c > 4 c/(1+ )
thus a sufficient condition for damped, monotonic behavior is:
2 > (1+ )c > 4 c/(1+ )
so it must be that 1 > 2 /(1+ ) or 1+ > 2 , or simply:
1 > 2 -  = 
Thus, a sufficient condition for damped monotonic stability is that  < 1. Thus, in Figure 1, we can
divide the area above the curve c = 4 /(1+ )2 into two parts. In that on the left (where  < 1 in area
I), we have a stable, monotonic equilibrium. In that on the right (where  > 1 in area III), we have
an unstable, monotonic equilibrium. At the point where c = 1 and  = 1 (implying repeated real
roots, at point II) we have a constant stationary state out of equilibrium.
Figure 1 - The Samuelson Oscillator
So far, so fine. What about the area with complex roots (i.e. below the curve c = 4 /(1+ )2)? Here,
obviously, D < 0. However, using imaginary systems, we know that the "modulus" of the system is
1. If the real parts of the roots are greater than 1, then the system has explosive oscillations whereas
if it is less than 1, then the system has damped oscillations. Now, the modulus is  (a2 + b2) = 1
where a = (1+ )c/2 and b =  (-D)/2. Now:
a2 + b2 = ((1+ )c)2/4 + [4 c - ((1+ )c)2]/4
or simply:
a2 + b2 =  c
Now, as the relevant modulus is 1 so that 1 =  (a2 + b2) = a2 + b2, then for damped oscillations, it
must be that:
1>c
and for explosive oscillations:
1<c
Obviously, we can draw the curve c = 1/ which forms the boundary between explosive and
damped oscillations in the case of complex roots. This is done in Figure 1. Obviously, this is an
asymptotic function which begins at c = 1 and  = 1. If c > 1/ (we are above the curve in area IV),
then we have explosive oscillations. If c < 1/ , which is below the curve (in area VI), then we have
damped oscillations. On the curve itself (area V), we have constant (harmonic) oscillations.
All the parameter regimes are thus drawn in Figure 1. They are as follows:
I - Damped monotonic
II - Constant monotonic
III - Explosive monotonic
IV - Explosive oscillations
V - Constant oscillations
VI - Damped oscillations
How does this multiplier-accelerator reveal "cycles"? In fact, only parameter ranges ( , c) which
are in situation E (constant oscillations) will yield constant cycles. All other parameter
constellations will result in something else -- either complete stability or complete instability (whether monotonic or oscillating). Thus, regular cycles are "structurally unstable" in the sense that
they emerge only if there is a precise parameter constellation and any slight movement or
displacement of the economy from these parameter values will end the regular cycle dynamics and
enter into either explosive or damped oscillations. Thus, the Samuelson multiplier-accelerator
model, as a model of the cycle, is incomplete as a theory of regular "cycles" as such, but a great
advance in the theory of "macrodynamics" and fluctuations in general.
We will leave for a future paper the continuation of our description of models imbedding cyclic
behavior done by Hicks, Dussenbery, Kalecki, Kaldor, Goodwin, etc. that either introduce
limitation such as ceilings and floors or nonlinear components that allow the occurrence of cycles.
Oscillatory behavior
We will go on now to propose a different approach based on the fact that the data series of GDP and
its components show oscillations. Our approach is based on the fact that the determination of the
solutions of the associated differential equations to the oscillatory behavior mentioned above may
be determined from the Fourier transform generated characteristic frequencies of the oscillations
and to discuss the behavior of the system in the associated complex space representation of the
solutions that indicates the attractors and the evolution of the system trajectories. This approach
allows the description of a dynamical system (for the moment of linear differential equations) which
later on we will analyze in terms of nonlinearities and potential chaotic behavior.
The specific conditions of human economic activity involve a cyclic behavior that is also evident in
the evolution of GDP. The various component sectors that generate value added are each having
their own cycles e.g. agriculture is obviously different from industry or from constructions but, they
all combine to generate GDP.
In Purica, 2010 and Purica and Caraiani, 2009, we have shown that the shocks in the economy are
triggering a typical exponentially amortized oscillatory response, e.g. in industrial activity, that
allowed the determination of the specific coefficients of the second order differential equation
describing the process.
The basic formulae for the above are given below:
A
0.18

1.52
d=
0.138701662
=
1.515393705
Td
45.3
1/a
130
- parameters from fitted data above
Td= 45.3 months
(years d = 3.775)
EXP(-ax)*A*SIN(2*/Td*x+)
- characteristic function
where:
n=A*d
 =atan(d/a)
d=2*/Td
i=a/n
undumped natural frequency
phase
dumped natural frequency
dumping ratio
Deduction of the values of interest is given below:
a=i*n
d=n*SQRT(1-i^2)
a=d(i/(1-i^2))
i=SQRT(1/(1+(d/a)^2))
 i=
0.055374283
n=a/i = 0.138914804
Tn= 45.23049473 months
(years n = 3.769207894)
And the differential equation results as:
d2y
dy
 0.015  0.019 y  0.019u
2
dt
dt
Figure 1. Industrial production response after 1990 shock.
Industrial Production
0.20000
0.15000
0.10000
0.05000
0.00000
Forecast N
-0.05000 1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166
cycle ind prod
-0.10000
-0.15000
-0.20000
-0.25000
Source Purica 2010
The fact that the specific equation for the industrial activity was determined is also giving the
possibility to apply this behavior to a later shock, such as the economic crisis at the end of 2008 in
Romania and to assess duration and amplitude of the economic response (industrial activity). Such
response is given in the figure below.
Figure 2. Evolution of industrial production from Jan.2009 - estimated
Source Author calculations
One may see that the down trend is reaching a first minimum after 22,8 months meaning almost the
end of 2010, after which the industrial production starts growing again. Considering the amplitude
and knowing that the decrease in GDP (we assume the same in industrial production) in 2009 was
7,1% and that this represents a -1 in normalized value of y above, then the decrease in 2010 will be
of aprox. 0,8 x 7,1% = 5,7% related to the value of 2008 (i.e. 6.11% if related to the previous year,
2009). An increase is expected in 2011 and it should be noticed that other oscillations will impact
the economy later on. Obviously the values calculated above are indicative and refer to industrial
production only. Any decision taken and implemented that changes the basic conditions is liable to
change the above trends either upward or downward.
The basic result here is the fact that there is predictive power in this approach and it is worth trying
to extend it to other components of the GDP.
Oscillatory behavior of the GDP and its components
To analyze this we have considered the evolution of GDP on a Quarter disaggregation for 20002008. Figure 3 is showing this trend for Agr(iculture), Ind(ustry), C(ons)tr(uction), Fin(ance) and
Serv(ices).
These components combine into the GDP and the analysis, we are starting with this paper, is due to
take into consideration each component and the way they combine into the GDP. For the moment
we remain with the industrial production.
Figure 3. Oscillatory behavior of GDP components
Source INS (National Institute of Statistic data plotted by the Author).
Let us give now a full example of the procedure we use to analyze the cyclic process.
Considering the case of the industrial production we are determining the Fourier transform of the
cyclic process. This requires to only consider 32 out of the 36 data points representing the quarterly
values of the GDP and its components. Based on these data we compute the Fourier transforms for
the resulting 32 frequencies and represent the amplitudes (in absolute value) in a frequencyamplitude space. The result is given in the figure 4 below.
Figure 4. Industrial Production Frequency – Amplitude representation of the Fourier transform.
Source Author calculation
Repeating the procedure for other GDP components we get their specific representations in the
frequency – amplitude space. We have grouped all the spectra into one graph but one may see the
variations in the spectra for various GDP component (see Figure 5).
Figure 5. GDP components’ spectra
Source Author calculation
Having done the above we are now in the position to go one step forward and discuss the type of
behavior described by the differential equation associated with the components. We consider that
the spectra above contain the eigen values of the associated process behavior. Since this paper is
opening the way to a comprehensive analysis we will present only the example of the industrial
production, the rest of the components will follow in further work.
Stability and complex space analysis
In order to assess the behavior of the given component we first apply the Bode plot such as to detect
the existence of a characteristic frequency resulting from the phase frequency and amplitude
frequency plots. (see also Schaum, 2003).
We consider the industrial production system described by the differential equation presented above
and write it in the time domain (s) then determine the transfer function of the system. This results
in:
((-0.019)/(s*(s^2+0.015*s+0.019)))
The Bode plot for the above is given below:
Figure 6.a) Amplitude Frequency plot
Figure 6.b) Phase Frequency plot
Source Author calculations
One may see that the maximum in amplitude is occurring for a given frequency which is the normal
frequency of the industrial production cycles (0,8632/2/=0,1389=n). While the phase is also
having a change at around the same frequency range.
Looking at the complex plane containing the solution represented by the Fourier transform
representation (Elmore and Heald, 1985) the roots of the equation (associated to the characteristic
(eigen) frequencies (Figure 7) show small values close to zero that we may neglect for now and the
remaining ones are one real and two complex conjugates all in the negative real axis. We will call
them 1, 2 and 3 with
3<Re 1,2<0
This configuration is specific for a rotation followed by a slower contraction toward the origin.
Let’s analyze now the complex space representation of the roots.
Figure 7.a. complex plane for the industrial production
Source Author calculation
We have chosen in the above the data series of the industrial production to give an example of the
way the analysis can be made both in the ‘phase space’ and in the time domain of the component.
The complex plane representation showed that the evolution trajectory is rotating with a slow
convergence toward the origin, which suggests a dynamically stable trend (see also Arnold 1974).
Figure 7.b. trajectories with oscillatory behavior in the phase space
We will not discuss here the other small points close to zero but leave their potential influence for a
later paper. We mention though, that the trajectories associated with the points considered (further
away from zero) will decay the first in the transient oscillations leaving the trajectories associated to
the closer to zero (negative and positive) points. These ones are the ones that persist on long term
and will potentially determine small amplitude potentially nonlinear behavior.
Conclusions
Considering the data series of the industrial production we have made some forecasting of its
evolution after the shock at the end of 2008 based on the second order differential equation
determined from the previous shock of 1990. The basic result was the upturn of the trend at the end
of 2010 leading to an expected increase in 2011. Further on we have extended the analysis based on
the data of the GDP and its components on a quarterly basis. We have calculated the Fourier
transform and the associated spectra for the industrial production as well as the other GDP
components. Moreover, based on the complex space representation we have used the Bode plot and
the roots in the complex space to assess the level of dynamical stability of the industrial production
evolution trajectory. The results show a rotational movement with a contraction toward the origin
that suggests the existence of a dynamically stable behavior. This enhances the amortized behavior
described by the differential equation determined previously. The paper opens the way toward a
thorough analysis of each GDP component and the GDP itself, in the framework of the oscillatory
dynamics presented here. The spectra above are definitely suggesting different behaviors and
combining them into the GDP may result in oscillatory behavior of this last one that has not been
modeled in this way till now. This contributes to increasing the variety of models available to make
predictions on economic evolution.
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