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	A Kaldorian Model A model conforming to Kaldor's specifcation was built using the hyperbolic tangent for the investment function in the form where A, ε and α are parameters and m and c are constants. The rate at which output changes to the level of capital is controlled by α which is generally greater than 1. This function was chosen because it is easily and instinctively manipulated to generate different investment profiles that might be suitable for modelling different economies. The constants m and c are chosen to keep the function in the positive quadrant. For simplicity a linear savings function was chosen where β < 1 represents the sensitivity of savings to capital. Model behaviour The system was simulated using MATLAB and Maple to determine its behaviour over a range of different parameter values. Values which gave cyclical behaviour were established using numerical techniques to generate a series of bifurcation diagrams. These values became a 'standard' set for producing reliable cycles and were individually varied to assess their effect on the sensitivity of the model. A phase portrait of the model is shown below which includes two trajectories and the accompanying time series. The units for the time series are presented simply as 'periods' which in this example is approximately twelve units. This can be scaled so that if a unit is considered as one accounting quarter, we would say that this represents, approximately, a four year cycle. Typical phase portrait and time series for model Bifurcation diagrams were produced by varying individual parameters to assess the behaviour of the model while relating parameter changes to the experience of real world economies. Increasing the value of parameter A stretches the investment function vertically so that small rises in income generate proportionately larger changes in investment plans. It can be considered as representative of the optimism among entrepreneurs, who require little incentive to take on new projects, or alternatively as an indication of the potential of emerging markets and new technologies. Parameter A can also be used in conjunction with ε to describe a variety of investment functions. The bifurcation diagram for A sees cycles generated between the values of 0.6 and 1.5 at which point they cut off abruptly. First impressions were that a small subcritical region might be present but closer inspection of the region at higher resolution failed to find any evidence. Modifications were made to the code so that the period of the oscillations might be studied. The results are shown below and reveal the period increasing exponentially as we approach the critical value. Bifurcation diagram and period with A as parameter This behaviour is appears to be consistent with a an infinite period bifurcation. The cause is likely to be a 'bottleneck' in the system which slows the trajectory in one part of the cycle until it becomes infinite. In the time series this can manifest itself as 'jerkiness' in the oscillation. Phase portraits and time series immediately before and after bifurcation at A ≈ 1.5 Further discussion of bifurcations can be found in chapter 6 of the report. Model Stability As an exercise the model was subjected to varying degrees of random noise to get an idea of its overall stability. Noise was added at varying levels to investment and savings functions individually to represent uncertainty in both areas. The model shows a good degree of resilience and maintains its underlying rhythm despite the interference. The example shown includes random elements of up to four times the equilibrium value. Resilience of time series to noise In the second image large amounts of noise were applied to a weakly oscillating system and after some initial upset (a large 'shock') the cycle is essentially restored. This is a modest example, but it perhaps illustrates how underlying cyclical behaviour, determined by simple nonlinear structures, may be inherent and even ubiquitous in an economy, despite persistently high levels of uncertainty.