The Limits of the
ECONOMIC CYCLE
an exercise in nonlinear economic dynamic modelling

by Andrew Rowson, supervised by Prof. A.R.Champneys
Home A Dismal Science Economic TheoryClassical Neoclassical Keynesian Cycles and Crises Measuring cycles Economic Time Series Data Processing Spectral Analysis Cycle Modelling Kaldor's Trade Cycle A Kaldorian model Conclusion Full Report	Economic Cycle Modelling Multiplier-accelerator models Early attempts at creating models that could generate cycles were based on the interaction between the multiplier and the accelerator. The accelerator principle describes how investment decisions are dependent upon expectations of future increases in demand, which in turn are based on past increases in aggregate demand or output. It can be expressed as a difference equation with investment It and output Yt at time t such that It = υ(Yt - Yt-1) where 0 ≤ υ < 1 is the 'accelerator coefficient' and represents the extent to which a producer responds to changes in demand. The multiplier is a Keynesian construct which describes how investment generates an increase in output. The marginal propensity to consume, c, is defined as ΔC = cΔY where proportionate changes in consumption and output are given by ΔC and ΔY and 0 ≤ c ≤  1. Any rise in output consists of a change in consumption and a change in investment such that ΔY = ΔC + ΔI. Combining these relations two gives ΔY = (1/1-c)ΔI  where (1/1-c) is known as the multiplier. The implication of these two relations is that investment will increase output (because of the multiplier) and increases in output will induce investment (by the accelerator). The concept was employed successfully by Paul Samuelson in 1939 in a model now known as the 'Samuelson oscillator' which successfully produced stable cycles but only along a boundary line in parameter space between two regions. John Hicks produced a modified multiplier-accelerator model of the trade cycle in 1950 which was also produced cyclic behaviour. Like the Samuelson model, cycles were only produced in a boundary case but the Hicks model was stable for one parameter value. Beyond this, explosive oscillations occurred, much as they had done for Samuelson. Hicks however argued for a natural 'floor' to disinvestment and a corresponding 'ceiling' for investment. The explosive behaviour was then effectively constrained so that stable oscillations proliferated between the two bounds. Unfortunately a constant rate of growth was a condition on the model and without growth the cycles simply decayed.