A Linear Model of Cyclical Growth

Abstract

PROFESSOR SAMUELSON's path-breaking article on Interaction Between Multiplier Analysis and Principle of Acceleration appeared in this REVIEW almost twenty years ago. A large literature has developed in which basic ideas of that article have been applied to both business cycle and economic growth problems. In a considerable portion of that literature, Samuelson's warning that the representation is strictly a marginal analysis to be applied to study of small oscillations has been overlooked.' Samuelson's warning can be interpreted as meaning that time series generated by any particular solution of model will determine actual income for only a short time. Given mathematical model, relevant particular solution can change due either to (i) accelerator or multiplier coefficients changing (as Samuelson suggests), or (2) imposition of new initial conditions. Goodwin2 has examined various models in which accelerator coefficient is a variable. These non-linear models are mathematically complex, and specific limit cycles that Goodwin derives obviously are due to special assumptions he makes about how path of income affects accelerator coefficient. Hicks 3 has investigated how an otherwise explosive accelerator model will be affected by floors and ceilings. In this paper such floors and ceilings will be interpreted as imposing new initial conditions, and therefore this paper can be considered a reinterpretation of Hicks's setup.4 We will work with a slightly modified version of Samuelson's model, and assume that

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