Relevance of Chaos and Strange Attractors in the Samuelson-Hicks Oscillator
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In this paper, we look for the relevance of chaos in the well-known Hicks-Samuelson’s oscillator model investigating the endogenous fluctuations of the national income between two limits: full employment income and under-employment income. We compute the Lyapunov exponent, via Monte-Carlo simulations, to detect chaos in the evolution of the income between the both limits. In case of positive Lyapunov exponent and large values of parameter (i.e. marginal propensity to consume and technical coefficient for capital), the evolution of income is seen to be chaotic. The model also may contain a quasi-periodic attractor that can be chaotic or not.
Let me, first of all, put the subject in context.
Up to 1930, crises were mainly considered by mainstream economic literature as an adjustment process to correct existing distortions in the economy. It was a sort of punishment for previous sins.
With Keynes’s General Theory, crises started to be considered as an economic illness requiring treatment.
Keynes devotes Chapter 22 of his famous book to the analysis of economic fluctuations. He followed Wicksell in modelling economic dynamic as the interaction of two rates of return. One of the two rates is the money interest rate. The other one is the marginal efficiency of capital. There are only marginal references to the role of the multiplier and the accelerator in the business cycle. It was Harrod who incorporated the interaction between them in a theory of economic fluctuations in an article he published in 1936, in parallel with the appearance of the General Theory.
However, when Samuelson published in 1939 his multiplier-accelerator model he did not mention Harrod but Hansen as the source of inspiration behind his contribution. Samuelson´s paper showed how cycles could be generated depending on the magnitude of the coefficients. However, as Verne rightly points out, this model is not able to produce lasting business cycles.
In 1950, Hicks publishes his book on the trade cycle. He adopts Samuelson´s model but he introduces an upper moving ceiling to the growth of production at the full employment level. When the rate of growth reaches the ceiling, for some values of the coefficients, this leads to a downturn.
The downswing is governed by the fact that income decrease leads investment to become negative. To avoid the extreme where no worn-out capital is replaced at all, Hicks introduced a floor at the depreciation level.
In his very interesting contribution, Verne shows that the Samuelson-Hicks model displays chaos for plausible and widely used parameter values. He shows the evolution of income between ceiling and floor for several values concerning the technical coefficient for capital and the marginal propensity to consume. For some of these values there appears a chaotic behaviour, being the coefficient for capital the key parameter in it.
Verne´s paper illustrates how different kind of behaviour appear depending on the numeric values of the parameters.
However, the main difficulty to introduce chaotic models in economic theory is that up to now very little empirical support for the presence of chaotic behaviour in economics has been found.
In principle, this should not be an obstacle if we take into consideration that economic data provide little evidence -if any- of linear, simple dynamics, and of lasting convergence to stationary states or regular cyclical behaviour; however, the linear approach absolutely dominates mainstream economics.
But what is accepted for “normal” science – in the Kuhnian sense – is not for its challengers. The detection of chaos in economic time series still faces some difficulties, mainly the fact that in economics we deal with short and time series as well as the high dimension of economic systems.
Anyway, chaotic behaviour may be or may not be relevant in economics. But what is beyond any doubt is that nonlinearity is by far a much more compatible assumption with most of the economic time series behaviour than linearity. As I have argued elsewhere [Beker, V.A . ( 2014). Why should economics give a new chance to chaos theory. In Faggini, M. and Parziale, A. (ed.) Complexity in Economics: Cutting Edge Research. Springer], nonlinearity is relevant in economics because it paves the way to the study of cyclic, non-periodic and chaotic behaviour.
This allows to study economic fluctuations as endogenously generated, contrary to the prevailing view in mainstream economics where fluctuations appear as the product of exogenous shocks. In line with the dominant equilibrium approach in economic thought, in the absence of such shocks the system would tend to a steady state, as different versions of the neoclassical model of optimal growth predict.
Reply from the author: First, I would like to sincerely thank Pr. Victor Beker for his comment. Beker is right to put in historical context the subject of the relevance of chaos in the Samuelson-Hicks oscillator.
It is true that traditionally, according to the neoclassical mainstream, economic fluctuations are seen to be the consequence of exogenous shocks, called also random perturbations, occurring in all economic activity. Thus, the forces of the market enable to resorb the disequilibrium resulting from such exogenous shocks. So, the influence of a random shock is only transitory, and the economic activity recovers its long-run trend quickly, which is characteristic of a non-significant hysteresis effect. Finally, according to this approach and more particularly the Real Business Cycle (RBC) theory, the fluctuations of the economic activity (traditionally measured by the rate of GDP growth) follow a trend stationary process.
However, in the thirties, Samuelson contradicts this vision by estimating that economic fluctuations are endogenous to the economic system. In his famous paper of 1936, he shows that, according to certain values of the coefficient for capital and marginal propensity to consume, the revenue can oscillate around its long-run equilibrium value. More later, in the fifties, Hicks improves the Samuelson model by adding the rate of growth to the variables as well as the ceiling and floor forming a corridor inside or, sometimes, outside (when the fluctuations are explosives if the coefficient for capital is above a certain threshold) which the revenue fluctuates and can show a chaotic evolution resulting in a strange attractor.
In fact, like the RBC theory, economic fluctuations exist around a long-run trend inside a corridor, but these are not issued from exogenous shocks but from the evolution of the global demand (that encompasses consumption and investment plus public investment) which can become chaotic in my opinion. Besides, as Beker points out, “nonlinearity is relevant in economics because it paves the way to the study of cyclic, non-periodic and chaotic behavior”. This is absolutely what I wanted to demonstrate in my paper by indicating that the chaotic dynamic may occur even in an endogenous fluctuations model.
Nevertheless, from an empirical point of view, the research of chaos is not easy because time series in economic are short, often non-stationary, and exhibit a non-linear evolution. As a matter of fact, there are some difficulties to detect chaos in economic series as Beker notices, but this is not impossible. In my recent contribution (Economics Bulletin, 2019), regarding the evolution of the Lebanese GDP growth rate during the 1970-2019 period, I put in light the chaotic dynamic of this variable (leading to a strange attractor) by carrying out the Brock, Dechert, Scheinkman (BDS) test (1996), on the one hand, and applying the Lorenz (1960, 1972) system of equations including GDP growth, consumption, and investment as variables, on the second hand.
Brock, W.A., Scheinkman, J.A., Dechert, W.D., Le Baron, B. (1996) “A test for independence
based on the correlation dimension” Econometric Reviews 15 (3), 197-235.
Lorenz, E.N. (1960) “Maximum simplification of the dynamic equations” Tellus 12, 243–254.
Lorenz, E.N. (1972) “Predictability: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” 139th Annual Meeting of the American Association for the Advancement of Science (29 Dec. 1972)”, in Essence of Chaos (1995), Appendix 1, 181.
Verne, J.F., Verne, C. (2019) “Chaos in Lebanese GDP: The Lorenz attractor approach”, Economics Bulletin, Vol 19, Issue 3, pp. 1958-1967.
In this paper, the author aims to show the existence of periodic orbits and a-periodic orbits in the original Samuelson – Hicks multiplier accelerator model. The author uses the Lyapunov exponent method and uses Monte-Carlo simulation method to detect periodic and aperiodic orbits in the model. In the original Samuleson’ (1939) model, in the parameter space between the marginal propensity to consume (say, c) and acceleration coefficient (say, k), Samuelson showed by varying the acceleration coefficient increased from 0 to 4 and at a particular value of the marginal propensity to consume the possibilities of a variety of dynamics in output. In particular, there was monotonic convergence, oscillatory converge, oscillatory diverge and monotonic divergence as on increased the value of k from 0 to 1 and beyond. Similarly, in Hicks’ (1950) model, in which Hicks rewrites the investment equation, the induced part of Samuelson’s model as a piecewise-linear equation and another piecewise-linear equation for the income ceiling, and shows damped cycles when k1. Goodwin (1951) modified the Hicksian ceiling and floor in as being determined by the net investment, and consequently it is the relation with the desired level of capital stock that generates oscillatory dynamics in his model.
In more recent times, with the hindsight of developments in dynamical-system theory, the issue of generating sustained oscillations in the class of multiplier-accelerator models was reconsidered by [Hommes, 1995], [Gallegati et al., 2003] and [Puu et al., 2005]. Hommes revisits this issue from the point of view of periodicity in the dynamic behavior generated by Hicks’ model. More precisely, Hommes poses the question whether every time path in Hicks’ trade cycle model converge to a periodic time path. He extends Hick’s model by considering lags in consumption and/or investment being distributed over several time periods (precisely three time periods) and demonstrates the existence of quasi-periodic and strange attractors. From the perspective of the underlying dynamics of the multiplier-accelerator model, Hommes’ extension of Hick’s trade-cycle model it is interesting that the attractors in the model exhibit periodic behaviour interspersed by sudden bursts of erratic behaviour, which is pertinent for understanding regime shifts that we encounter in real economies. In a more detailed investigation of the dynamics of Hicks’ model, [Gallegati et al., 2003] analyse bifurcations to study the conditions under which the model produces periodic and quasi-periodic dynamics.
In the context of the rich and diverse literature on this subject, which is briefly discussed above, I have the following suggestions for the author to consider while revising the paper,
(i) A brief review of literature that includes more recent work in this area would be useful for the reader to have a perspective of the literature as well as it will provide a context for the current paper.
(ii) Second, the author must make it clear what model is under investigation. In p. 13, the author makes the point that “The assumption that investment depends on the lagged change in output is not essential; the accelerator effect also arises if investment depends on the current change in output. But in that case, chaos does not arise as output is a first-order difference equation, not second-order.” While the above statement is correct, but then it is not immediately clear to me for which model the author is applying the Lyapunov exponent method from the statement that follows, viz., “Thus, if output is a second-order equation, the occurrence and relevance of chaos, measured by the Lyapunov exponent, depend on the values of capital coefficient (k) and marginal propensity to consume (c)”. The above statement does not make it clear, what 2 is the model under investigation. It would be useful if the author discusses the model for which the Lyapunov method is used and how that model is different from what has been done in the literature.
(iii) It is not clear to me whether there is any economic motivation for applying the Lyapunov method and Monte-Carlo simulations. Is it just a mathematical curiosum of determining aperiodic fluctuations and strange attractors, or there is a compelling economic objective for the application of the method? I think it would be useful for the author to consider those readers who may not be well-versed in the details of the dynamical systems methods but may be interested in the underlying economic question that motivates a particular mathematical method.
Gallegati, M., Gardini, L., Puu, T., & Sushko, I. . Hicks’ trade cycle revisited: cycles and bifurcations. Mathematics and Computers in Simulation, 63(6), 505-527.
Goodwin, R.  “The nonlinear accelerator and the persistence of business cycles,” Econometrica 19, 1–17. In, Hicks, J.  A Contribution to the theory of trade cycle, Clarendon Press: Oxford.
Hommes, C.  “A reconsideration of hicks’ non-linear trade cycle model,” Structural Change and Economic Dynamics 6, 435–459.
Puu, T., Gardini, L. & Sushko, I.  “Hicksian multiplier-accelerator model with floor determined by capital stock,” Journal of Economic Behaviour and Organization 56, 331–348.
Samuelson, P.  “Interactions between the multiplier analysis and the principle of acceleration,” The Review of Economic Statistics 20, 75–78.
Reply from the author: First, I would like to sincerely thank Pr. Raghavendra for his very constructive comment.
The literature advised by S. Raghavendra was very useful to improve my paper. Thus, the three suggested references such as Hommes (1995), Gallegati et al. (2003), and Puu et al. (2005) have been included in a new subsection as a brief literature review. The paper of Piiroinen and Raghavendra (2019) has also been considered for providing a context for the paper.
Regarding the model for which the Lyapunov exponent is applied, I explain in Section 4 (with a more precise title), that the Lyapunov exponent can be seen as one of the most relevant tools for showing the occurrence of chaos in dynamical systems as well as in time series regarding the economic or financial data. The originality of the paper, with respect to the existent literature, consists in using such a tool for detecting the occurrence of chaos in the Samuelson-Hicks model. Besides, it is true that my assertion “Thus, if the output is a second-order equation, the occurrence and relevance of chaos, measured by the Lyapunov exponent, depend on the values of capital coefficient (k) and marginal propensity to consume (c)”, is somehow confusing. So, I have corrected this inaccuracy by indicating that if the output is a second-order equation, the occurrence and relevance of chaos can be measured by the Lyapunov exponent which is a useful tool for exhibiting the national income trajectory between the floor and ceiling. Such a trajectory of national income depends on the values of capital coefficient (k) and marginal propensity to consume (c).
About the economic motivation for applying the Lyapunov method and Monte-Carlo simulation, I explain, at the beginning of Section 4 of the revised paper, that I carry out a Monte-Carlo simulation to analyze the national income behavior which depends on the simulated values of the coefficient for capital, k. I precise the purpose by mentioning that this method enables us mathematically to determine the aperiodic fluctuations and strange attractors in the Samuelson-Hicks model. I also indicate that using the Lyapunov method allows describing the trajectory of a macroeconomic variable but not specifically to reach an economic objective.