ON CHAOS : THEORY AND APPLICATIONS

Abstract

The mathematical discovery of chaotic dynamical systems has initiated a renewed interest in nonlinear dynamics, which do not simply constitute some kind of a generalization of known linear systems, but which indeed concern the very conceptual framework of an understanding of actual phenomena. Over the past years, chaos has been applied to diverse areas of economics, such as in models of growth cycles, in cobweb models of demand and supply as well as in models of cournot competition. By applying the chaos theory, economists are able to describe complex phenomena ranging from steady equilibrium to disequilibrium and from stable cycles to aperiodic cycles, which are difficult or sometimes impossible to model with the traditional static analysis and linear modelling techniques.

Although the possibilities of chaos in overlapping generations models of pure exchange have been previously discussed by many economists, the studies on models relevant to the Singapore economy have never been attempted. Our goal is to fill this gap by studying an overlapping generations model that incorporates social security. Here we aim to illustrate that the economy need not necessarily be in equilibrium in the long run and that chaos may exist over a selective range of parameters.

In particular, the two-period model in our study is based on Hoon's (1991) model. Hoon associates the two different periods with the active working 'young' and the retired 'old'. However we modify this by further distinguishing between the working and non-working within each if the two groups. Individuals in the second period arc said to cam a higher income since they have been working for a longer period and arc thus more experienced. Eventually, the standard techniques of nonlinear analysis are applied before comparative static analysis can be performed. Our results demonstrates three different types of trajectory paths: convergent, divergent and chaos. This differs from Hoon's model where the economy will always be in long-run equilibrium.

The model is then extended to the analysis of period-doubling. This involves adjusting the rate of time preference so that a stable two-period cycle evolves into a four-period one and then a eight-period one, which will eventually result in an infinite cycle length. This bifurcation procedure, can repeat itself until a certain value is reached so that the time path will exhibit a random-like behaviour. A discussion about the sensitive dependence on initial conditions follows. This means that time paths of initial points that are close together become incomparably unlike. Therefore, difficulties in long tern forecasting is experienced as a result of these properties of chaos.

Subsequently, we show that the government can stabilise the economy by increasing the employer's social security rate when chaos is present. Since increasing the social security rate causes an increase in the slope of the phase diagram at equilibrium, our analysis incorporates a higher social security rate so as to achieve an absolute value of less than one for the slope. This is an important characteristic of a stabilise economy. In fact, similar results can be achieved when the share of wages to national income is increased. However the minimal control that government welds over this parameter limits policy options to that of varying the social security rate.

In addition to the theoretical concepts of chaos discussed earlier, the problems of chaos in economics arc highlighted in the final chapter. Recommendations for future research are then put forward as an apt conclusion.