The Elements of a Nonlinear Theory of Economic Dynamics

The inter-war years produced some of the richest theories in the history of economic thought and it was for this reason that Shackle (1967) entitled his now classical work on theoretical developments in that era “The Years of High Theory”. The period saw in particular the development of the theories of Hayek, Keynes, Harrod, Kalecki, The Stockholm School and Schumpeter which aimed at explaining various aspects of the dynamic evolution of the capitalist economy. The 1930s culminated with Samuelson’s “Foundations of Economic Analysis”, even though this work was not published until 1947. The original 1941 version carried the subtitle “The Operational Significance of Economic Theory”s and was a major landmark in theoretical economic analysis as it provided a framework in which the particular problems and viewpoints of the earlier mentioned theorists could be handled in a unified general way.

Our aim in this chapter is to expound those aspects of the theory of dynamical systems which shall be most relevant to our later investigations. Our approach will be discursive in that we shall try to paint a broad brush picture of the concepts and techniques of the modern qualitative geometric view of the theory of dynamical systems. We have already outlined in chapter one the revolution in the approach to the analysis of dynamical systems which has occured in recent years. Our exposition is inspired by the works of Arnold (1973) and Hirsch and Smale (1974), which were the first texts to diffuse the modern concepts to a broad mathematical audience, and the recent text of Guckenheimer and Holmes (1983), which is a very readable account of some of the most recent developments such as strange attractors and chaotic behaviour. Whilst we do not prove any theorems and our exposition places much emphasis on geometric arguments, as the liberal sprinkling of diagrams will bear witness, our discussion does become a little more technical when we expound two of the major tools of analysis of nonlinear dynamical systems namely the method of averaging and the method of relaxation oscillations.

In chapter one we mentioned that there are two broad approaches to business cycle theory — the endogenous cycle theories and the exogenous shock theories. In this chapter we shall use the concepts and techniques of the previous chapter to place a class of endogenous cycle theories into a unified mathematical framework. In particular we shall be considering theories whose basis is in the dynamics of the real sector. This emphasis in no way indicates a belief that these theories are the unique or most satisfactory explanation of the business cycle. Our choice is dictated partly by the fact that these theories provide a well recognized body of economic knowledge which can benefit from a unified viewpoint and partly by the fact that they are still seen as the most representative of the endogenous cycle theories.

In the last chapter we demonstrated the two basic tools for the analysis of limit cycle solutions to nonlinear differential equations, viz. the method of averaging and the method of relaxation oscillations, on some of the traditional endogenous business cycle theories. We saw there that this approach allows us to view such theories within a unified mathematical framework, and we were thereby able to make a number of simple extensions to these models. We were also able to gauge the effects of traditional countercyclical policies on the amplitude of the cycle.

In the last two chapters we have applied some of the nonlinear techniques of chapter two to various macroeconomic models which mathematically reduce to a system of two nonlinear differential equations. In this chapter we shall illustrate the analysis of a three dimensional nonlinear system by use of the centre manifold concepts discussed in section 2.7. A convenient model on which to illustrate these concepts is Goodwin’s (1967) model of cyclical growth. This model for some considerable time received scant attention in the economics literature, apart from the contribution of Desai (1973). However recent interest in theories of economic cycles has led to a resurgence of activity on Goodwin’s model, see e.g. Goodwin et al., (1984), Blatt (1983), van der Ploeg (1983), Medio (1979), and Vellupillai (1979).

In chapters three, four and five we have seen various examples of limit cycle motion in dynamic economic models. However as we pointed out in section 2.7 limit cycles are not the only type of oscillatory motion which can arise in dynamical systems. For differential equation models of dimension three or greater chaotic motion is possible and indeed is more likely to be the norm. In the last chapter we speculated that as the time lag increases the limit cycle motion of the Goodwin model would evolve into chaotic motion, probably via the process of period doubling.

In this chapter we bring to bear a number of the concepts and techniques used in earlier chapters on an important problem in the modern theory of economic dynamics. This is the so called dynamic instability problem.

In chapter one we stated that the main aim of our thesis was to routinize the use of nonlinear methods in dynamic economic analysis. We achieved this aim by making use of the method of averaging and the method of discontinuous or relaxation oscillations to investigate nonlinear models which can be reduced to a set of nonlinear differential equations on the plane having limit cycle solutions. Higher dimensional models may be investigated by use of the concepts of centre manifold theory, which concentrates attention on limit cycle motion on an appropriate two dimensional manifold. The method of averaging usually yields a differential equation containing two elements. The first consists of the linearised part considered in the traditional local linear analysis. The second is a term which captures the qualitative effects of the nonlinearities in the model. The equilibrium of this differential equation approximates the amplitude of the limit cycle motion of the nonlinear model; by considering the effect of parameter shifts on this equilibrium we are able to determine the qualitative effects of parameter shifts on the amplitude of the cycle. In effect the qualitative analysis of the limit cycle equilibrium has been reduced to a comparative static type calculation since the principal achievement of the method of averaging is to reduce the analysis of a two dimensional limit cycle to that of a point equilibrium. The possibility of doing a comparative dynamic analysis in this way is an advantage of the approach we propose compared to those approaches referred to earlier in the thesis which rely on the use of existence theory of nonlinear differential equations and which cannot do much more than tell us that a limit cycle exists. One of the major criticisms of the use of limit cycles to model economic fluctuations is that the fluctuations are too regular but the discovery of the concept of chaotic behaviour overcomes this criticism — as we have seen such behaviour can arise in one dimensional difference equation models or in differential equation models of order at least three. We believe that the tools we have provided should allow the economic theorist to finally break free of the straitjacket of local linear analysis which has prevented a proper analysis of many important issues in dynamic economic theory. The economic theorist will henceforth be able to concentrate on questions such as when is a nonlinear analysis required, what are the important nonlinearities and how best to model these.