# Nonlinear Cycle Theory

Zarnowitz (1985, pp. 540 and 544) argues that nonlinearities are likely to be very common in economic relationships. Business cycle models incorporating nonlinearities can generate limit cycles, which can be regarded as the equilibrium motion of the economy, whereas linear models cannot. Limit cycles occur when the energy dissipated over a period is compensated for endogenously so that there is neither a gain nor a loss of energy and a steady oscillatory state results. There may be equilibrium points or growth paths in nonlinear models that have limit cycle solutions, but these are unstable and the representative point will tend to follow the cycle instead. Limit cycle solutions need not be unique and may be stable or unstable. The limit cycle followed will, however, be independent of initial conditions. In the stable limit cycle case, the effect of adding shocks will be to increase irregularity by causing temporary deviations from the equilibrium orbit, which itself need not be symmetric (see also section 1.3). The sine wave produced by linear models is a conservative oscillator in the sense that no energy is dissipated. The cycle followed by the representative point in the conservative oscillator case will depend on the initial conditions. When hit by shocks, however, the conservative property leads to an explosive motion and consequently the practical usefulness of conservative models, which are also symmetric in linear formulations, is doubtful - unless a Hicksian approach is to be adopted and ceilings and floors imposed. But this entails dropping the original linearity assumption.

This accounts for the adoption of the linear Frisch-Slutsky approach, in which damped motions occur because energy is dissipated over time but new compensatory energy is supplied by exogenous shocks, as an alternative. The existence of a stable limit cycle implies that the economy will always be gravitating towards an endogenously determined cyclical motion. Nonlinear models with such solutions can be regarded as modern attempts to derive an endogenous theory of the cycle. They are therefore related to previous attempts to develop endogenous theories of the business cycle, which are reviewed briefly by Zarnowitz (1985, section III), and they contrast with the mainstream linear Frisch-Slutsky models, which rely on external shocks to supply energy and have dominated the post-war business cycle literature.

Some of the recent contributions to the literature on nonlinear business cycles will be reviewed in this section, which aims to update the survey in Mullineux (1984, section 2.5). It should be noted that economists working in this area often try to develop models of dynamic economic development, in which cycles and growth are part of the same macrodynamic process. Some of the ideas introduced in this section will be considered further in Chapter 4.

Chiarella (1986) analyses a model with a nonlinear demand for money function which is dependent on the expected rate of inflation. Initially the money market is allowed to adjust sluggishly with inflation expectations being formed adaptively. Chiarella shows that the model has a stable limit cycle if the expectations time lag is sufficiently small. By allowing the time lag to go to zero, perfect foresight is considered as a limiting case. It is found that the stable limit cycle persists. Apart from providing an additional demonstration16 that plausible nonlinear functions can result in a stable limit cycle, Chiarella is able to cast light on the dynamic instability problem that arises because perfect foresight, in the sense that rate of change variables such as inflation can be correctly perceived, leads to saddle point instability. The introduction of a nonlinearity removed the instability commonly associated with linear perfect foresight models by replacing the unstable local saddle point equilibrium with a global stable limit cycle equilibrium.

Papers presented at an international symposium on nonlinear models of fluctuating growth17 were published in Goodwin el al. (1984) and included a number of further extensions18 to the Goodwin (1967) growth cycle model as well as other contributions to the theory of fluctuating growth. The symposium rejected, as analytically unsatisfactory, the simple superposition of fluctuations on growth trends because, it was argued, fluctuations and growth interact in a crucial way. Instead it advocated that a general theory of fluctuating growth be pursued. Such issues will be discussed further in Chapter 4. A central view of the symposium was that economic fluctuations are a natural, endogenously determined, consequence of the internal dynamic structures and conflicts inherent in capitalist economies and that advanced capitalist economies undergo fluctuations whether or not there is state intervention. Linear models were rejected as inadequate and judged incapable of representing the complex relationships inherent in capitalist economies.

The symposium judged the two keystones to an understanding of why capitalist economies evolve cyclically as having been provided by Marx, who stressed class conflict, and Schumpeter, who emphasised the role of technical progress. Goodwin's (1967) growth cycle model brought these two elements together using a model, drawn from biology, of the symbiotic relationship between predator and prey populations, with capitalists as predators and workers as prey, and spawned a series of studies on Marx/Goodwin cycles. These are listed in Goodwin el ah (1984). In Goodwin's seminal paper, technical progress was introduced as a trend and steady growth of the labour supply was also assumed. Growth was introduced via two log linear trends and was not fully integrated with the cyclical motion around it.

Desai (1973) extended the model by introducing inflation and expected inflation and by allowing the capital-labour ratio, assumed constant by Goodwin, to vary over the cycle as utilisation rates, proxied by employment, changed. The effect of introducing inflation is to complicate the wage bargaining process and has a stabilizing influence unless workers are able to incorporate actual wage inflation into their wage demands. Desai and Shah (1981) further extend the model by reformulating the technical change relationship. They incorporate the Kennedy-Weizacker technical change function (discussed in Samuelson 1965) and find that the introduction of induced technical progress changes the stability properties of the model. The conservative oscillations in Goodwin's original model are the result of the implicit assumption that each side in the class struggle has only one weapon. Workers can bargain for wages, their bargaining power being dependent on the level of employment, and the capitalists can determine the growth of employment by their investment decisions. Desai and Shah's model gives the capitalists an additional weapon: the choice of the induced rate of technical progress. As a consequence, an equilibrium point, rather than a conservative cyclical motion, results.

Van der Ploeg's (1984) contribution to the symposium was to consider the effect of introducing endogenous technical progress, based on Kaldor's (1957) technical progress function, and allowing workers to save and to receive dividends from share ownership. Goodwin (1967) had assumed that workers consumed all their income. The implication of the analysis is that the class conflict, and with it the cycle, is likely to die away as workers obtain an interest in capitalism.

Di Matteo (1984) considers the implications of introducing money into the Goodwin model and this enables him to examine the interplay between real and monetary factors. Two cases are examined. In the first, the money supply is assumed to be exogenously determined and it is found that the share of profits is inversely related to the rate of growth of the money supply. If it can control the money supply, the central bank can have a profound effect on the cycle and can in fact adopt a rule to eliminate it if certain initial conditions are satisfied. In the second case, it is assumed that the central bank sets the interest rate rather than the money supply and again, if certain initial conditions are satisfied, the central bank can adopt a rule to eliminate the cycle. Di Matteo stresses, however, that the analysis is highly abstract since it incorporates no theory of the behaviour of the banking sector. Such a theory is necessary to facilitate an analysis of the interplay between financial and industrial capitalists in the Schumpeterian tradition (see section 4.2).

The symposium includes other interesting extensions of Goodwin's model. Glombowski and Kniger (1984) examine the effects of introducing unemployment benefits while Balducci et al. (1984) use the theory of non-co-operative differential games to explain the effects of introducing the REH into the model. Balducci et al. find that the cycle remains under the REH. This indicates that the cycle is due not to myopia but to the fundamental conflicts inherent in economic development under capitalism which the model tries to capture. In an attempt to move away from the high level of aggregation which he now finds unsatisfactory, Goodwin's contribution was to analyse economic interactions within the framework of multi-sectoral models. The multi-sectoral approach has subsequently been developed by Goodwin and will be discussed further in section 4.5.19

Another significant contribution to the nonlinear cycle theory literature is that of Grandmont (1985), who used nonlinearities to generate an endogenous EBC. Unlike the Lucasian EBC and RBC models, it required no shocks to keep it alive and, in contrast to New Classical models, money turned out to be non-neutral. This was despite the fact that as in other EBC models, markets clear, in the Walrasian sense, at every date and in addition traders have perfect foresight. The latter can be regarded as rational expectations with full information, in contrast with the imperfect information imposed by the "market islands' hypothesis employed in Lucasian EBC models. The equilibrium output is shown to be negatively related to the equilibrium level of the real rate of interest, and the employment of the nominal rate of interest as an instrument of monetary policy is shown to be extremely effective. A simple deterministic counter-cyclical policy can enable the monetary authorities to stabilize business cycles and force the economy back to a unique stationary state.

The source of Grandmont's endogenous deterministic cycles is the potential conflict between the wealth and intertemporal substitution effects, which are associated with real interest rate movements. Business cycles emerge when the degree of concavity of the traders' utility functions is sufficiently higher for older than younger agents. This follows from the assumption that older agents have a higher marginal propensity to consume leisure within the simple structure of an overlapping generations model. Grandmont's analysis implies that cycles of different periods will typically coexist. He feels his results suggest that economic theorists should look more closely at the sort of mechanisms that might be responsible for significant nonlinearities in the economic system if they wish to have a proper foundation on which to build a sound business cycle theory. He postulates that relaxation of the ad hoc Walrasian continuous market clearing assumption and the introduction of imperfect competition may lead to the sort of nonlinearities that give rise to endogenous economic fluctuations. It would also reduce reliance on variations in real interest rates through variations in relative prices by allowing quantity adjustment and thus mechanisms akin to multiplier-accelerator effects to play a role. A sound Keynesian, or non-Walrasian, business cycle theory could then be developed. This, he argues, might form the basis of the 'New Keynesian' business cycle research programme, referred to in the introductory section of this chapter, although most contributions20 have so far tended to conform to the linear Frisch-Slutsky approach.

To conclude this section, the contributions of Day (1982) and Varian (1979) are discussed. Varian (1979) employs catastrophe theory to examine a variant of the nonlinear Kaldor (1940) model.21 Kaldor's model includes sigmoid savings and investment functions that intersect in a manner that generates a stable limit cycle solution (see Chang and Smyth 1970). Catastrophe theory was developed by Thorn (1975) (see also Zeeman 1977) to describe biological processes and has since been widely applied. Catastrophe theoretic models entail a system of differential equations in which the parameters are not constant but change at a much slower rate than the state variables. There are, therefore, essentially two sets of variables. The 'fast, or state, variables can be regarded as adjusting towards a short-run equilibrium and the 'slow' variables, or parameters, as adjusting in accordance with some long-term process. Catastrophe theory, therefore, studies the movement of short-term equilibria as long-run variables evolve and would appear to be a particularly useful tool for business cycle analysis and the study of dynamic economic development. When a short-run equilibrium jumps from one region of the state space to another, a catastrophe is said to occur. Catastrophes have been classified into a small number of qualitative types, the simplest of which is the 'fold catastrophe'. This occurs when the system contains one "slow' variable and one 'fast' variable. For a given value of the slow variable, the fast variable adjusts to a stable equilibrium. If the state space contains 'bifurcation points' at which there are abrupt changes in stability characteristics, as in the Kaldor model, then adjustment to a locally stable equilibrium can involve jumps or catastrophes. Things naturally get more complicated as more fast and slow variables are added. With one fast variable and two slow variables, for example, "cusp catastrophes', which allow jumps and then either fast or slow returns to short-term equilibria, can occur.

Using a cusp catastrophe, Varian shows that if there is a small shock to one of the stock variables in the Kaldor model, a similar story to that analysed using the simpler fold catastrophe unfolds and an inventory recession of minor magnitude results. If the shock is relatively large, however, wealth may decline sufficiently to affect the propensity to save and a depression can result because the recovery can take a long time. This is related to the idea, discussed by Leijonhufved (1973),22 that economies operate as if there is a 'corridor of stability', within which small shocks are damped out but large shocks are amplified. Large deflationary shocks may, for example, produce financial crises (see Chapter 3) and waves of bankruptcies which throw a normally stable system into a deep depression. Varian suggests that catastrophe theory might usefully form the basis for some further business cycle research.

Day (1982) applies the mathematical theory of chaos which, like catastrophe theory, is related to bifurcation theory,23 to show that in the presence of nonlinearities and a production lag, the interaction of the propensity to save and the productivity of capital can generate growth cycles that exhibit a wandering, saw-toothed, pattern, not unlike observed aggregate economic time series. These 'chaotic' fluctuations need not converge to a cycle of regular periodicity and are not driven by random shocks. Periods of erratic cycling can be interspersed with periods of more or less stable growth. Under such circumstances, the future of the model solution cannot be anticipated from past realisations. A deterministic single equation model is found to be consistent with structural change and unpredictability. Such models allow periods of sustained growth, such as that experienced since the early 1980s, but suggest that recent claims that a combination of supply side initiatives and fine tuning have eliminated the cycle are likely to prove to be incorrect. Day's work indicates that even if there is substance to the Monte Carlo hypothesis (discussed in section 1.2) that there are no regular business cycles, economic fluctuations may still be a phenomenon to be reckoned with, and that random shocks may not be as important for driving cycles as the Frisch-Slutsky approach (discussed in section 1.4) indicates.

Goodwin has employed catastrophe theory and related ideas drawn from bifurcation theory for the analysis of dynamic economic development. He adopts a multi-sectoral approach which makes nonlinear analysis intractable and finds it necessary to make linear approximations, which hold in the short run, and to view the long run as a series of short runs. The linear approximations are chosen carefully, however, to generate regions of stability and instability between which economic variables can bifurcate back and forth; Goodwin also makes use of fast and slow variables.

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