(A) Kaldor's Non-Linear Cycle
(B) Kalecki's Distribution Cycle
(C) Goodwin's Non-Linear Accelerator
(D) Goodwin's Class-Struggle Model
The multiplier-accelerator structures reviewed above have linear dynamic structures. As a result, cycles are generated and maintained only by structurally unstable parameter values (Samuelson) or dampened dynamics with continuous exogenous shocks (Frisch-Slutsky) or exogenously-constrained explosive dynamics (Hicks). As a result, early Keynesian linear multiplier-accelerator fall dangerously close to an "untheoretical" explanation of the cycle -- precisely what the original Oxbridge research programme was designed to avoid.
However, linear structures are often adopted because they are simple and the results they yield are simple. But simplicity is sometimes more a vice than a virtue - particularly in the case of macrodynamics and economic fluctuations. It is not only unrealistic to assume linearity, but the very phenomena that we are out to uncover, the formation of cycles and fluctuations, becomes relegated to the "untheory" of exogenous shocks, ceilings, floors, etc. The contention of Lowe (1926) and many Keynesian writers is that theories of fluctuations ought really to explain how fluctuations arise endogenously from a working system otherwise (paraphrasing Lowe's title), how is business cycle theory possible at all?
As a result, many economists have insisted that non-linear structures should be employed instead. Why interest ourselves with non-linear dynamics? As one famous scientist answered, for the same reason we are interested in "non-elephant animals". In short, non-linear dynamical structures are clearly the more general and common case and restricting attention to linear structures not only unrealistically limits the scope of analysis, it also limits the type of dynamics that are possible.
Richard Goodwin (1951) was among the first to insist on the use of non-linear dynamical systems in business cycle theory to generate endogenous fluctuations. A decade earlier, Nicholas Kaldor (1940) had used non-linear investment and savings functions to generate trade cycles - without the assistance of mathematical formalism. Michal Kalecki (1935, 1937, 1939, 1954) alternated between using linear systems with shocks and non-linear systems proper. These models are not only formally different from the Harrod-Hicks linear multiplier-accelerator models, but those of Kalecki and Goodwin, in particular, rely on heterogeneous classes of agents and income distribution dynamics, ideas which owe perhaps more to Marxian or structural cycle-and-growth tradition thought than to Keynes's General Theory directly. They have, nonetheless, emerged as a distinct strand of Post Keynesian thought.
Beyond the fact that these pioneers of non-linear dynamic systems in economics were all Keynesians, there is a natural fellowship between non-linear dynamics and Keynesian cycle theory: namely, that non-linear systems, in contrast to linear systems, are far more capable of yielding regular endogenous macrofluctuations. This implies that fluctuations are the outcome and indeed an integral part of a working economy. As this concept was central to Keynes's (1936) static theory, it is no surprise that Oxbridge researchers insist on endogenous cycles as a way of extending it into a dynamic context. In contrast, Neoclassical theory seems to be more apt to consider equilibrium as opposed to fluctuations as the central feature of a working economy -- and thus prefer to conceive of "fluctuations" as the results of erratic displacements or aberrations from a working economy. The Neoclassical concept of the economy, thus, is perfectly compatible with linear systems; the Keynesian concept of endogenous cycles, however, seems to require non-linear structures.