Mathematical Models of Investment Cycles

Abstract

We propose and investigate three mathematical models describing production cycles. They incorporate various mechanisms of endogenous fluctuations in economic systems. The models are based on ODE systems. The first model is a Keynesian IS-LM model of business cycles. The interest rate is determined by the money market and influences the relationship between savings and investments, allowing funds to flow from one to the other and vice versa. In the second case, the fluctuation mechanism is associated with time lags between investment growth, capital growth, and rate of return on capital. As a result, the economy periodically “overheats”, as rapid growth of capital suppresses the return rates, production becomes unprofitable, and investments sharply decline. Two models realizing this mechanism are proposed. One is a minimalist model based on a system of three ODEs. The other is an augmented model that sufficiently fully describes modern economic systems of developed countries and consists of nine ODEs and nine algebraic equations. It encompasses all the principal markets: labor market, capital market, financial market, and commodity market. Bifurcation analysis of the three models is carried out, oscillation regions are determined, and oscillation mechanisms are examined in detail. The model parameters are chosen so that the cycle periods are 12–17 years long.