Phase-Locking in a Goodwin Model

Abstract

Mode-locking is a phenomenon that arise in nonlinear dynamical systems, when two or more cyclic motions interact with one-another. In the first place mode-locking means that the oscillations adjust to each other so as to get in step. However, the adjustment is more complicated than this because each mode contains harmonics and subharmonics which may also try to get in step. Many economic phenomena can be thought of as coupled oscillators. Examples are the interaction between seasonal fluctuations and the business cycle, the interaction between different sectors in an economy (Lorenz, 1987a), and the interaction between different national or regional economies (Lorenz, 1987b).We consider the simplest case of coupled oscillators, namely a model that consists of an independent oscillator which acts as a forcing on an internally generated cyclic motion. The internal motion is produced by a Goodwin model of economic business cycles (Goodwin, 1951). Perhaps, the Goodwin model does not meet the standards of business cycle models of today. However, it is easy to illustrate mode-locking with this model, and mode-locking can arise in more popular models as well through the same channels as illustrated here. Mode-locking has at least one important empirical implication:If a model with mode-locking is estimated by standard econometric techniques, the obtained parameter values will typically be in error because the observed behaviour is influenced in a complicated manner by the nonlinear interaction between the cycles. This has important consequences for policy formulation.