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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. 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