Non-resonating cycles in a dynamic model for investment behavior

Schumpeter (1939) conjectured that there exist many, or perhaps even an indefinite number of subcycles in an economy, but that in the long run only a few subcycles remain stable and persist, whereas others become unstable and cease. In this paper, we build upon this idea, hypothesizing that, for subcycles to persist in a multiple cycle structure, their cyclical lengths should not be close multiples. We formalize our theory using a mathematical model for investment behavior, inspired by Austrian business cycle theory. The model is quasi-periodic and satisfies the conditions of Kolmogorov–Arnold–Moser theory. This enables us to describe the dynamic properties of the model. We conclude that the cycles of the system should be non-resonant for the motions to remain stable in the long run. That is, that the ratios of the cycle lengths should be sufficiently irrational. Coupling this with the recent empirical findings of De Groot et al. (2021), who measured the ratios between subcyle lengths in GDP in 25 OECD countries and found them to be close to the golden ratio φ, we conclude that the detected cycles therein are non-resonant and remain stable.