# Relevance of Chaos and Strange Attractors in the Samuelson-Hicks Oscillator

## Abstract

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.