Non-linear Endogenous Non-stochastic Aggregate Fluctuations (NENAF)

And that was what I was interested in—trying to formulate a model which would show that Harrod was right and Tinbergen wrong in that rather brutal review he wrote of Harrod’s book ... I was concerned for ten years with trying to formulate a model which would show that Harrod was right in his intuition, however preposterous his inept formulation—and it was this almost singleminded pursuit, which finally led me to the formulation of the one-sided oscillator (Goodwin to Velupillai, 17 June 1985; italics added).

The non-linear, endogenous, non-stochastic theory of macrodynamic fluctuations did not emerge out of a theoretical vacuum; instead, it came out almost as a natural consequence, given the direction in which business cycle theory was developing in the early 1930s. But the publication of The General Theory (Keynes 1936 [1973]) and The Trade Cycle by Harrod (1936) were the immediate reasons for Goodwin’s 10-year odyssey mentioned above. Harrod (ibid.) sought to combine the Keynes-Kahn multiplier with the accelerator—he (Harrod) referred to is as ‘the relation’ (ibid.: 58)— together with assumptions on imperfect competition, resource constraints^[1] and relatively inflexible prices, and interest rates over the business cycle.^[2]

For selecting these features, akin to what Kaldor (1957) later came to call the stylized facts (albeit in the context of aggregate growth theory), one appealed to reasonably reliable empirical evidence and broad patterns during those turbulent times. These chosen characteristics had to be theoretically substantiated and historically congruent while at the same time encapsulating economic data that was available. Aggregate fluctuations in capitalism’s dynamic contours^{[3] [4]} were synthesized and summarized, in retrospect, as the following:

• 1. Evidence for the persistence of economic fluctuations;
• 2. The asymmetric nature of economic cycles (i.e. the observed time series of aggregate variables were significantly non-sinusoidal);
• 3. Presence of multiple equilibria;
• 4. Instability of equilibria, at least, locally; and
• 5. Presence of significant non-linearities in the relationship between economic variables and/or in the behaviour of the aggregate variables defining the macroeconomic fluctuations.

These five essential features—persistence, asymmetry, multiple equilibria, instability, and non-linearity—which were chosen as the modelling criteria, in turn, implied an endogenous cycle. The orthodox view of the economy was tied to the idea that it was in a state of equilibrium and stable. Departures from this equilibrium, if any, were eliminated by self-adjusting mechanisms of the market forces. The above desiderata—both in terms of the economic hypotheses that they resorted to (non-linearities in relationships, multiple equilibria, instability) and the dynamic contours they relied on and attempted to explain (persistent and asymmetric fluctuations)—also highlight a marked departure from the prevailing orthodox views of the functioning of the economic system in the advanced industrial economies. In this, Goodwin adhered squarely to the Cambridge Keynesian tradition of Kahn-Sraffa-Robinson-Kaldor, but he emphasized, almost always, a growth cycle aspect, to which was added the Schumpeterian element of innovations. This spanned his contribution to NENAF theory from 1946 to 1967 (Goodwin 1946, 1967)—15 of these 21 years were spent at a predominantly Keynesian Cambridge; the earlier years at Harvard were very much Schumpeterian, but infused with copious elements of Keynes and Harrod in Goodwin’s contributions to business cycle theory (the locus classicus of which was Goodwin 1951a).

It was in Goodwin (1951a) that the first formal, mathematical model encapsulated all the five desiderata mentioned above; but to these were added an Occamist principle of minimal non-linearities—essentially, one. His insistence on non-linearity as the key and a fruitful way of explaining maintained oscillations was explicit not only from the title of this classic but also by the observation that ‘economists will be led, as natural scientists have been led, to seek in nonlinearities an explanation of the maintenance of oscillation. Advice to this effect, given by Professor Le Corbeiller in one of the earliest issues of this journal, has gone largely unheeded’ (Goodwin 1951a: 2; italics added).

The results reported in this classic were first presented at the December 1948 meeting of the Econometric Society in Cleveland and summarized in the April issue of Econometrica in 1949 (Goodwin 1949b) before the full version was published in 1951.^[5] Harrod’s book on the trade cycle claimed that a combination of accelerator and multiplier was the cause of dynamic instability in the Keynesian system. But it was criticized by Tinbergen (1937), pointing out that the first- order system, as Harrod conceived it, is incapable of generating cycles.^[6] Goodwin believed in Harrod’s insight and finally found a way of formalizing it by resorting to non-linearities. His excursion into the world of non-linear models came about through the perseverance^[7] and the decisive influence of Le Corbeiller, who, as noted, personally tutored Goodwin on the non-linear theory of oscillations. In his own recollection of how he came towards building his model, Goodwin says:

Both Frisch and Tinbergen had failed to pay adequate attention to a short note in an early issue of Econometrica to the effect that to explain a self-generated cycle, it was necessary to have an unstable equilibrium with a pair of nonlinearities, in the outer regions of the state space, to convert the instability into global stability. During the war I had the accidental good fortune to work in the same laboratory as the author of the note, Phillipe Le Corbeiller, and he taught me what I needed to know about maintained oscillation (Goodwin 1988: 20; see also footnote 4 above).

In Hicks (1950), exogenous autonomous growth factors are superimposed on to a disequilibrium model of fluctuations.^[8] Unlike this, or the shock theories which superimposed exogenous, random shocks to an equilibrium model of growth, Goodwin took an alternative path. He realized that the ‘the inherent instability [of] the accelerator principle could be dampened by the multiplier mechanism in a model in which the equilibrium was locally unstable but globally stable’ (Velupillai 1998: 1,439). To achieve this, he combined the dynamic multiplier with a non-linear accelerator and weaved it together with Schumpeter’s theory of how innovations from within the system drive growth. The ‘steady evolution of ideas leads to intermittent, irregular, but not random, bursts of expenditure’ (Goodwin 1951a: 3; italics added) and it supplied the energy to the system for the ‘maintenance of oscillation’. The famous Goodwin (1951a, 1967) models have become part of the folklore of mathematical macrodynamics and are very much a part of textbook exposition (the clearest of which, in my opinion, is Gandolfo 1981). Instead, I want to emphasize a different aspect of Goodwin’s Occamist credo.

Over a research and teaching career spanning more than 60 years, Goodwin relied on the following systems of non-linear, ordinary, differential equations for his contributions to NENAF theory:^[9]

• 1. The van der Pol equation: x -т(1 - x² + x = 0
• 2. The generalized forced van der Pol equation: x + f (x, x)x + g (x') = 0(t)
• 3. The Goodwin-Le Corbeiller equation:

x + p(-exp (x)-2 )x + x = 0 0