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	Kaldor's Trade Cycle Model A prototype for nonlinear models The model of the business cycle presented by Nicholas Kaldor in 1940 has been referred to as a 'prototype' model for nonlinear economic dynamics because it succeeds where the previous models fail and generates endogenous cycles without the need for shocks, time lags or heavily prescribed parameter values. Furthermore it was derived from existing theories. Kaldor concluded that functions for savings and investment cannot both be linear. From Keynes, economic output tends towards a level where savings and investment are equal, thus if investment exceeds saving, there will be a positive adjustment in output and an excess of saving over investment results in a contraction of output. If savings, S, and investment, I, are both linear functions of output, Y , then there is one equilibrium point and only two situations are possible as can be seen below. Equilibrium with linear saving and investment functions In the first instance, dI/dY >  dS/dY so that above the equilibrium level of output I > S and output grows, while below equilibrium, S > I and output contracts. The equilibrium is unstable and leads either to unfettered growth or the total collapse of the economy. In the second case, dI/dY < dS/dY and a reverse argument shows that the equilibrium is unconditionally stable. Needless to say, neither option reflects actual experience. Introducing nonlinear functions Kaldor proposed instead a sigmoidal investment function, where dI/dS is small at extreme levels of output. At low output levels there is likely to be excess capacity in the economy and increases in demand can be met without the need for significant investment. At high levels of output, increasing costs, and the fact that most of the 'best' opportunities will already have been taken, act as a disincentive to further investment. A similar argument can be used to justify a corresponding nonlinear savings function, but in this example a linear savings function is retained for simplicity. The nonlinear function introduces the possibility of two additional equilibria which can be seen below. Additional equilibria with nonlinear investment function Cycles arise by considering the effect of accumulating capital or the stock of goods, K which can be used in the production of others. Increasing capital is associated with falling investment and savings, with investment being the more responsive of the two. The Kaldor model can be expressed in general terms as I = I(Y,K) and S = S(Y,K) and the rate of change of output depends on the difference between the two so that where the constant λ is a measure of the speed of adjustment. The rate of change of capital is, by definition, investment although a depreciation term can also be subtracted for completeness to give Together these equations describe the Kaldor model.