A nonlinear dynamic model of short run fluctuations

Most recent attempts to model the business (see the work of Lucas (1975), Sargent-Wallace (1975), Taylor (1979)) adopt a Frisch-Slutsky framework, in which the persistence of fluctuations in macroeconomic activity is explained by the persistence of random shocks impinging on an otherwise stable (and usually linear) systematic component. Work by Kaldor (1940) and Goodwin (1951), as well as Hicks (1950) (and more recently Ichimura (1954), Klein and Preston (1969), Change and Smyth (1971), Torre (1977) and Varian (1979)) attempted to model the persistence of fluctuations (or what was then believed to be an endogenous cycle) with systematic economic behaviour by employing nonlinear behavioural functions. The nonlinear dynamic approach to modelling cycles has used either a variant of the Poincare-Bendixon theorem, to prove the existence of closed orbits, or a theorem on Leinard's equation (a second order non-linear differential equation) to prove the existence, uniqueness and stability of limit cycles. This paper shows that the output dynamic of a modified version of a traditional, dynamic IS-LM macro-model is reducible to Leinard's equation. The models considered here are modifications of a traditional IS-LM model in two important ways. First, we augment the model with a government budget constraint allowing for money and bond financing; and secondly and more importantly, we model investment behaviour as a Kaldor-type nonlinear function of output. It is shown that for cases in which the general equilibrium of the model is locally unstable, the model is nevertheless globally stable in the sense that all locally unstable points converge to a unique and stable limit cycle. The paper proceeds as follows. Section 2 sets up the general model, and discusses its geometric properties. Section 3 derives Leinard's equation and proves the existence, uniqueness and stability of a limit cycle for various substructures of the model. Section 4 draws some implications. The Appendix states a theorem on Leinard's equation used in the text.