Nonlinear mode-interaction in the macroeconomy

A central problem for a dynamic formulation of macroeconomic theory is how to explain the occurrence of different, relatively well-defined economic modes such as the short term business cycle, the construction (or Kuznets) cycle, and the economic long wave (or Kondratiev cycle). Equally important is a description of the various phenomena that can arise through interaction between these cycles. Modern nonlinear theory suggests that different cyclical modes may be entrained through the process of mode-locking, where the periods of the interacting modes adjust to one another, so as to attain a rational ratio. This type of interaction is well documented in physical and biological systems. However, despite the importance of the problem and abundant evidence for nonlinearity in the economy, modern concepts of nonlinear mode-interaction have not yet been applied to the problem of entrainment between economic cycles. We show how mode-locking and other highly nonlinear dynamic phenomena arise in a model of the economic long wave. The behavior of the model is mapped as a function of the frequency and amplitude of an external forcing, producing both a devil's staircase and a detailed Arnol'd tongue diagram. Two different routes to chaos are identified. The Lyapunov exponents are calculated, allowing the strength of the chaos to be assessed, and the fractal nature of the basins of attraction for two simultaneously existing periodic solutions is illustrated. The paper concludes with a discussion of the implications for economic theory.