(A) The Samuelson Oscillator
(B) Metzler Inventory Cycles
(C) Hicks's Trade Cycle
(D) Duesenberry-Smithies Ratchet Effects
(E) Growth Cycles: Duesenberry-Pasinetti
(F) Extensions: Shocks and Money
Roy F. Harrod laid out the basic tenets of the Oxbridge research programme in his 1936 The Trade Cycle: An essay -- much of it written before Harrod even saw a draft of J.M. Keynes's General Theory. Harrod contended "that by a study of the interconnexions between the Multiplier and the Relation the secret of the trade cycle may be revealed" (Harrod, 1936: p.70). This "Relation" was the acceleration principle of investment.
One ought to note that J.M. Keynes himself did not have much credence in a deterministic accelerator as employed by Harrod and the Oxbridge models. Instead, Keynes had argued that it was expectations dynamics that generated cycles by affecting the marginal efficiency of investment and subsequently the multiplier and output (Keynes, 1936: Ch.22). Nevertheless, Keynes left the topic undetailed. Thus, Roy Harrod went on alone, in his theory of the trade cycle (1936) and later on in his theory of growth (1939, 1948), to explore the relationships between the Keynesian multiplier and accelerator-type investment functions to explain a growing, progressive economy with and without cycles.
The principle of the multiplier, as laid out by R.F. Kahn (1931) and J.M. Keynes (1936) is that if investment increases, there will be an increase in output as a result of a "multiplier" relationship between equilibrium output and the autonomous components of spending, in this case:
DY = DI/(1-c)
where c is the marginal propensity to consume, Y is output and I is investment. The principle of the accelerator, as laid out by Albert Aftalion (1913) and John Maurice Clark (1917), was that investment decisions on the part of firms are at least in part dependent upon expectations of future increases in demand, which may, in turn, be extrapolated from any current or past increases in aggregate demand or output, e.g.
It = b (Yt - Yt-1)
Thus, the multiplier principle implies that investment increases output whereas the acceleration principle implies that increases in output will themselves induce increases in investment.
Consequently, it would at least seem natural if some bright economist put these two together and examined the dynamic properties of investment and output as they affect each other, perhaps in generating cycles and/or growth. The first such bright economist was Roy F. Harrod (1936), albeit his analysis was purely verbal and not without some knots. His associated attempt to formalize a Keynesian growth model (Harrod's, 1939, 1948) was not much more successful: he ended up with his famous "knife-edge" instability.
Harrod's fellow Oxford economist, John Hicks (1949, 1950) picked up where Harrod left off. Hicks's (1950) trade cycle model sought to recast Harrod's unstable "multiplier-accelerator" dynamics into cyclical ones by having explosive trajectories bang up against floors and ceilings. To this end, Hicks employed the formalism of dynamical difference equations, that had been introduced in a similar context by Paul Samuelson (1939) in his income-expenditure "oscillator" and by Lloyd Metzler (1941) in his inventory cycle.
Hicks's (1950) "forced non-linearity" of ceilings and floors were somewhat restrictive. The essential Hicksian model was expanded upon by James Duesenberry (1949) and Arthur Smithies (1957) to include "ratchet effects" and thus obtain cycles along a output growth path. Later, Duesenberry (1958) and Luigi Pasinetti (1960) considered a different accelerator for the Hicksian model that would yield both growth and cycles. A bit more distinctly, Richard Goodwin's (1951) exercise added a non-linear accelerator to generate cycles endogenously while D.J. Smyth (1963) attempted to incorporate a monetary "LM" side to the standard multiplier-accelerator model. We shall review these "simple" multiplier-accelerator models before turning to endogenous cycle theory, where "natural" non-linearities are put in place to generate cycles and growth.